Compatibility conjecture for sequences satisfying the length condition

Let ${\widetilde{w}}=(i_1,\ldots,i_r)$ be a sequence of elements of the vertex set of a Dynkin quiver, put $J=[1,r]$, and define $j^+$, $J_e$, $J_f$, $w_{\leqslant t}$, the matrix $\widetilde{\mathsf{B}}^{\widetilde{w}}$, and the skew-symmetric matrix $\Lambda^{\widetilde{w}}$ as in the construction preceding the claim. Assume that, for every interval of the sequence of length at most the length of the longest Weyl-group element, the corresponding product of simple reflections has full length.

Compatibility conjecture. For any ${\widetilde{w}}$ satisfying this condition, the pair $(\Lambda^{\widetilde{w}},\widetilde{\mathsf{B}}^{\widetilde{w}})$ is compatible, namely

$$\Lambda^{\widetilde{w}}\widetilde{\mathsf{B}}^{\widetilde{w}}=(-2d_{i_s}\delta(s=t))_{s\in J,\;t\in J_e}.$$

This is presented as a generalization of an earlier compatibility proposition; the paper states that the conjecture is proved later under an additional condition, while the general case is not resolved in the supplied text.