I think finding combinatorial models for the Kronecker coefficients, or the multiplicative structure constants for Schubert polynomials would make good polymath projects.

These are quite famous problems in the field of algebraic cominatorics, and would immensely give better insight. Furthermore, the problems are quite accessible, it boils down to fit some combinatorial model to some known (but tricky to compute) data, meaning that even a bright high-school student can give it a try.

Both these problems has the Littlewood-Richardson coefficients as special cases, for which there are plenty of combinatorial models, so a good start would be to collect these, and see if there is some way to generalize these.

Both problems have known models for other sub-cases, so one can imagine that a polymath project could study natural sub-families. In the case of Schubert structure constants (indexed by three permutations), we know the answer for vexillary permutations.

Perhaps it is possible to find models for other natural combinations of permutations.

Finally, it might even be possible to do some "bruteforce" approach, by some kind of machine-learning. I have not heard this being used before, but it is not totally impossible that Schubert structure constants are given by lattice points in certain nice polytopes (since known special cases are), so it might be possible to try to find polytopes to the data.