Sigma action on denominator matrices for source-sink moves

Let $\mathbb{T}_n$ be the $n$-regular tree, let $t_0\xrightarrow{k}t_1$ be an edge, and let $B_1=\mu_k(B_0)$. Assume that all entries in row $k$ of $B_0$ weakly agree in sign. Let $\sigma_k$ be the piecewise-linear map defined on integer vectors by leaving all simple-root coordinates except the $k$th unchanged and sending the $k$th coordinate according to

$$[\sigma_k(\beta):\alpha_k]=-[\beta:\alpha_k]+\sum_{\ell=1}^n|(B_0)_{k\ell}|\bigl[[\beta:\alpha_\ell]\bigr]_+.$$

It acts on integer matrices columnwise.

Sigma denominator-matrix conjecture.

$$D_t^{B_1;t_1}=\sigma_kD_t^{B_0;t_0}.$$

This conjecture is described as closely related to the source-sink denominator recursion and uses the Cartan companion and its associated piecewise-linear Weyl-group modification. The source gives no resolution.