The period-determines-mutation conjecture for Fano polygons

Let $P_1$ and $P_2$ be Fano polygons. Their singularity content is the pair consisting of the sum of the local singularity-content integers and the basket of residual singularities. Let $L_{P_i}^T$ denote the affine spaces of maximally-mutable Laurent polynomials with Newton polygon $P_i$ and $T$-binomial edge coefficients, and let $\pi_{P_i}$ be their classical periods.

Period-determines-mutation conjecture. If $P_1$ and $P_2$ have the same singularity content and there is an affine-linear isomorphism $\varphi\colon L_{P_1}^T\to L_{P_2}^T$ such that

$$\pi_{P_1}(a,t)=\pi_{P_2}(\varphi(a),t),$$

then $P_2$ is obtained from $P_1$ by a chain of mutations.

The claim seeks to characterize mutation equivalence using singularity content and equality of classical periods. It is presented as one of two further conjectures concerning the relationship between combinatorial mutations and mirror-symmetry data.