Signed-column conjecture for denominator matrices

Let $B_0$ be an exchange matrix with initial vertex $t_0$ in the $n$-regular tree $\mathbb{T}_n$. A matrix has signed columns if every column has either all nonnegative entries or all nonpositive entries.

Signed-column conjecture. For all $t\in\mathbb{T}_n$, the matrix $D_t^{B_0;t_0}$ has signed columns.

This conjecture is quoted as a significant weakening of an earlier conjecture and is intended to relate the two source-sink conjectures. The source gives no resolution.