The k-Schur filtration conjecture

Let $k'>k$, let $\mu\in\mathcal{B}^k$ and $\lambda\in\mathcal{B}^{k'}$ be partitions, and let $s_\mu^{(k)}[X;t]$ denote the graded $k$-Schur functions.

The k-Schur filtration conjecture. There are polynomials $\tilde b_{\mu\lambda}^{(k\to k')}(t)\in\mathbb{Z}_{\geq 0}[t]$ such that

$$s_\mu^{(k)}[X;t] = \sum_{\lambda\in\mathcal{B}^{k'}} \tilde b_{\mu\lambda}^{(k\to k')}(t)\,s_\lambda^{(k')}[X;t].$$

This is the structured filtration of graded $k$-Schur functions that organizes their branching from level $k$ to level $k'$. The source gives no resolution status.