Coincidence of tagged-triangulation and cluster frieze patterns

Let $S_{\odot}$ be the punctured disc and let $\widetilde{\mathcal T}$ be a tagged triangulation of it. The constructions $F(\widetilde{\mathcal T})$ and $F_c(\widetilde{\mathcal T})$ are frieze patterns associated with $\widetilde{\mathcal T}$.

Coincidence conjecture. For every tagged triangulation $\widetilde{\mathcal T}$ of $S_{\odot}$, the frieze patterns coincide:

$$F(\widetilde{\mathcal T})=F_c(\widetilde{\mathcal T}).$$

The preceding theorem proves the assertion when the cluster corresponds to a slice, and the conjecture extends this agreement to every tagged triangulation of the punctured disc.