The Stokes triangulation–seed correspondence conjecture

Let $T$ be a labeled Stokes triangulation of an orbifold $(\mathbf{O},\mathbf{A})$. Consider a sequence of signed flips

$$T'=\mu_{k_N}^{(\varepsilon_N)}\circ\cdots\circ\mu_{k_1}^{(\varepsilon_1)}(T).$$

Let $(B,x,y)$ be a seed with $B=B(T)$, and apply the same sequence of signed mutations:

$$(B',x',y')=\mu_{k_N}^{(\varepsilon_N)}\circ\cdots\circ\mu_{k_1}^{(\varepsilon_1)}(B,x,y).$$

Stokes triangulation–seed correspondence conjecture. We have $T=T'$ if and only if $(B',x',y')=(B,x,y)$. This conjecture asserts that the seed obtained from signed mutations records exactly whether the corresponding sequence of signed flips returns the Stokes triangulation to its initial state.