The branching-path conjecture for k-Schur functions

Let $\mathcal C^{k+1}$ and $\mathcal C^k$ denote the sets of $(k+1)$-cores and $k$-cores, respectively. For $\lambda\in\mathcal C^{k+1}$ and $\mu\in\mathcal C^k$, let $b_{\mu\lambda}^{(k-1\to k)}(t)$ be the branching coefficient in the $k$-Schur expansion, and define

$$b_{\mu\lambda}^{(k)}(t):=\sum_{[\mathbf p]\in\overline{\mathcal P}^{k}(\lambda,\mu)}t^{{\rm ch}(\mathbf p)},$$

where $\overline{\mathcal P}^{k}(\lambda,\mu)$ is the relevant set of paths in the poset of $k$-shapes and ${\rm ch}(\mathbf p)$ is their charge.

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

$$b_{\mu\lambda}^{(k-1\to k)}(t)=b_{\mu\lambda}^{(k)}(t).$$

The conjecture identifies adjacent-level branching coefficients with charge-generating functions of paths in the poset of $k$-shapes. It was stated in earlier work and is known in the specialization $t=1$, but remains unproved in the stated graded form.