Cylindric loop Schur function conjecture for rigged-configuration shapes

Let $B=B^{1,s_1}\otimes\cdots\otimes B^{1,s_m}$ be a product of one-row crystals, let $p=b_1\otimes\cdots\otimes b_m\in B$, and define $x_j^{(i+j-1)}$ to be the number of occurrences of the letter $i$ in $b_{m+1-j}$. For $1\leq s\leq n-1$, let $\lambda(s,0)=(n-s)^m$, and recursively obtain $\lambda(s,r+1)$ by removing the largest ribbon strip containing the bottom row and having at most $n$ boxes. Let $\mathscr{D}_s(\lambda(s,r))$ be the periodic propagation of $\lambda(s,r)$ by shifting $n-s$ steps right and $s$ steps up. Cylindric loop Schur function conjecture. For every integer $r\geq 1$,

$$\operatorname{trop}\left(\frac{s_{\mathscr{D}_s(\lambda(s,r-1))}^{(0)}}{s_{\mathscr{D}_s(\lambda(s,r))}^{(0)}}\right)(x_j^{(i)})=\nu_r^{(s)},$$

where $\nu_r^{(s)}$ is the $r$-th part in the $s$-th shape of the rigged configuration $\Phi(p)$. This conjectures that the shapes of rigged configurations are obtained by tropicalizing ratios of cylindric loop Schur functions, extending the known relation between affine-crystal energy functions and tropicalized loop Schur functions; the status of the general assertion is not resolved in the supplied text.