The marked-surface quadrilateral bijection conjecture for X-seed patterns

Let $(S,M)$ be a marked surface and let $\mathcal{S}$ be an $\mathcal X$-seed pattern from $(S,M)$. For every triangulation $T$ of $(S,M)$ and every $\gamma\in T$, let $q_T(\gamma)$ denote the quadrilateral associated with $\gamma$, and let $\mathcal X(\mathcal{S}_{sf})$ denote the set of $\mathcal X$-variables of the corresponding seed pattern. Quadrilateral bijection conjecture. The map

$$\{q_T(\gamma)\cup\{\gamma\}\mid T\text{ a triangulation of }(S,M),\ \gamma\in T\}\longrightarrow\mathcal X(\mathcal{S}_{sf}),$$

which sends $q\cup\{\gamma\}$ to $x_{q,\gamma}$, is a bijection. This extends the quadrilateral description of $\mathcal X$-variables from the finite-type cases treated in the paper to seed patterns from arbitrary marked surfaces.