The charged branching-polynomial conjecture for k-Schur functions

Let $\Pi^k$ be the poset of $k$-shapes, let $\mathcal{C}^k$ be the set of $k$-cores, and let $\mathcal{P}^k(\lambda,\mu)$ be the paths in $\Pi^k$ from $\lambda$ to $\mu$. Write $\overline{\mathcal{P}}^k(\lambda,\mu)$ for their equivalence classes under the charge-preserving diamond relations, and let $b_{\mu\lambda}^{(k)}(t)$ be the branching polynomial.

Charged branching-polynomial conjecture. For all $\lambda\in\mathcal{C}^{k+1}$ and $\mu\in\mathcal{C}^k$,

$$b_{\mu\lambda}^{(k)}(t)=\sum_{[\mathbf{p}]\in\overline{\mathcal{P}}^k(\lambda,\mu)}t^{\operatorname{charge}(\mathbf{p})}.$$

The ungraded analogue, obtained by counting path-equivalence classes, is stated as a theorem immediately before this conjecture. The source gives no resolution status for the graded refinement.