The ILC could help address the matter-antimatter symmetry problem, neutrino oscillations and possible candidates for dark matter.

yes

See smaller particles.

Could help understand the cohesion of matter and anti matter worlds

Taking into account that ILC means “clean” collisions and the collision energy around 1TeV we could have much more clean Higgs bosons which should help us study its properties. Additionally we may see some SUSY particles

Because the matter-antimatter problem is solved by a particle of Hydrogen between them, removing could be done with proper accelerator energy. Since Dark Matter is composed of the universe of Hydrogen particles, removing them would cause the universe of matter to become non-existent.

ILC could help with supersymmetry, dark matter, potentially other Higgs bosons, understanding the neutrino mass. It won’t help us quantize gravity, though (in fact, I believe quantizing gravity is the wrong way to do it, we should gravitize quantum mechanics not quantize gravity).

Since mater/antimatter are determined by rotation/antirotating of the Hydrogen Particles I believe your statement is incomplete.

The proposed International Linear Collider (ILC) would attempt to study the following,

1. The Coupling of Higgs Boson with other particles: The Higgs mechanism is one of the areas in Particle Physics that must be studied in greater details to test the validity of the standard model. One of the properties of the Higgs Boson is its ability to couple with other particles. The aim of the ILC would be study the strengths of these couplings enabling us to know whether the predicted coupling strength of the Higgs Boson with other particles is correct. With the information that we receive from the ILC, we would develop an understanding of whether there is a single Higgs Boson or a family of Higgs Bosons imparting masses to a wide array of particles.

2. The self coupling of Higgs Boson: The standard model predicts that Higgs Boson couples with itself too with a very high precision. The ILC would try to understand the self coupling of the Higgs Boson understanding the potential of the coupling. This would help us confirm or reject whether the so called Higgs Boson is the one predicted by the Standard model.

3. Validating the Vacuum Stability of the standard model: The ILC would also try to validate the standard model at higher energy regimes assuming that the LHC does not find any deviations from the standard at its operating energy. This is done by studying the stability of the standard model also called its vacuum stability. This is done by studying the Higgs mass and the mass of the top quark, the later being extremely important because the stability of the theory depends on the mass of the Top Quark.

4. Search of Dark Matter Particles: At present Scientists have been trying hard to figure out the dark matter particles. If the dark matter particles have energies in TeV, ILC would have the capability to detect them and would also characterize their properties by precisely calculating the mass, spin and the parity of the dark matter particles.

5. Revealing Extra Dimensions: Modern theories like the superstring theory tries to explain gravity at the quantum scales by adding additional spatial dimensions to each point in space. These extra dimensions must be too small because we do not see them. We do not have any observations that shows its existence and this is where the ILC comes into picture. It is speculated that matter might be made up of particles that live in these extra dimensions. Such particles would have higher energies and therefore higher masses. Measurement of these masses might tell us how the extra dimensions look like. If the energies are in the TeV range, the ILC might discover these hidden dimensions by looking at extremely energetic collisions producing particles that disappear in the extra dimensions.

For a more comprehensive understanding, click here

Dark matter should be highest on list

Gaps in the standard model get filled when electron-protons are shot together and results reveal structures only theorized before.

Better colliders, higher energy, higher better knowledge simple.

1. Existence of Supersymmetry
2. Extra dimensions

Well, its design and proposed capabilities suggest it could address several gaps and questions in our understanding of particle physics. For instance,
Precision Measurements of Standard Model Parameters: The ILC is designed to provide high-precision measurements of the properties of known particles within the Standard Model. By precisely determining parameters such as mass, coupling constants, and decay rates, scientists can test the internal consistency of the Standard Model and look for deviations that could indicate new physics.
Higgs Boson Studies: The discovery of the Higgs boson at the LHC was a significant achievement, but there is much more to learn about this particle. The ILC could contribute by providing precise measurements of the Higgs boson’s properties, helping to understand its interactions with other particles and potentially revealing any deviations from the expected behavior.
Search for New Particles: While the LHC can produce particles at very high energies, the ILC’s design allows it to produce cleaner and more precise collisions at lower energies. This makes it well-suited for studying particles associated with potential new physics scenarios, such as supersymmetry or other beyond-the-Standard-model theories.
Dark Matter Candidates: Although the direct production of dark matter particles is challenging, the ILC could contribute to the search for dark matter indirectly by studying known particles that could be associated with dark matter. For example, if there are new particles in the dark sector that interact weakly with ordinary matter, the ILC could provide insights into these interactions.
Understanding Quark and Lepton Structure: The ILC’s design allows for precise studies of quark and lepton interactions. By probing the internal structure of these particles, scientists can gain a deeper understanding of the fundamental building blocks of matter.
Electroweak Symmetry Breaking: The mechanism of electroweak symmetry breaking, responsible for giving mass to particles, is a central aspect of the Standard Model. The ILC could contribute to a more detailed understanding of this process and help verify the Standard Model’s predictions in this area.
In summary, the International Linear Collider has the potential to significantly advance our understanding of particle physics by providing precise measurements and exploring energy regimes complementary to those accessible at the LHC.

I believe that there are many gaps in our scientific understanding of the standard model that may be addressed by the ILC.
Understanding fundamentally smaller particles is a significant prospect, we may be able to further study and understand the structure of quarks, as these are our fundamental blocks of normal, baryonic matter, this is of high importance.
Also, the real world of matter-antimatter relationships, new light has recently been shed on these relations, such as positron-electron reactions and relationships, this may be further studied.
Lastly, I believe that dark matter is one of the big new frontiers in modern physics, and the ILC may be able to work toward further studying and understanding what dark matter fundamentally is.

The gravitron perhaps?