Invariance of sign stability under mutation-loop representations

Let $b3_i:t_i\to t'_i$ $(i=1,2)$ be edge paths in the exchange graph $b4_I$ representing the same mutation loop $d\rm[b3_1]_{b2}=d\rm[b3_2]_{b2}$. A path is sign-stable when it has the sign-stability property defined for mutation paths. Invariance conjecture. The path $b3_1$ is sign-stable if and only if $b3_2$ is. This would show that sign stability is an invariant of the mutation loop rather than of its chosen representation sequence; the paper reports that many experiments support the claim, but no proof or counterexample is known.