Decomposition-independence conjecture for vineyard matching

Let $S=\{\sigma_1,\dots,\sigma_N\}$ be a filtered simplicial complex, and let $\tau\in\mathfrak S_N$ be a permutation of its simplices. Write

$$\tilde S=\{\sigma_{\tau(1)},\dots,\sigma_{\tau(N)}\}.$$

For any sequence $J=\{i_1,\dots,i_k\}$ satisfying

$$\tau=\prod_{j=1}^k(i_j\,i_j+1),$$

the sequence determines a vineyard matching between the reduced bases of $S$ and $\tilde S$. Decomposition-independence conjecture. This vineyard matching does not depend on the choice of the sequence $J$. This would make the vineyard matching canonical for an arbitrary simplex permutation, rather than dependent on its decomposition into adjacent transpositions; the source reports independence in experiments but leaves the general statement as a conjecture.