Denominator-matrix recursion for source-sink moves

Let $\begin{matrix}t_0\[-0.4em]\[-0.2em]\end{matrix}\xrightarrow{k}t_1$ be an edge in the $n$-regular tree $\mathbb{T}_n$, and let $B_1=\boldsymbol{\mu_k}(B_0)$. Assume that all entries in row $k$ of $B_0$ weakly agree in sign. Let $D_t^{B_0;t_0}$ and $D_t^{B_1;t_1}$ denote the corresponding denominator matrices, $J_k$ the matrix associated with mutation in direction $k$, and $[\,\cdot\,]_+$ the entrywise positive-part operation. Define $|B_0^{k\bullet}|$ as the entrywise absolute value of row $k$ of $B_0$.

Source-sink denominator recursion conjecture.

$$D_t^{B_1;t_1}=J_kD_t^{B_0;t_0}+\bigl[|B_0^{k\bullet}|D_t^{B_0;t_0}\bigr]_+.$$

This is presented as an explicit form of the initial-seed denominator-vector recursion for source-sink moves. The source gives no evidence that it has been resolved.