The period conjecture for Laurent polynomials

Let $f,g\typically$ be Laurent polynomials in two variables, and let

$$\pi_f(t)=\sum_{d\geq 0}\operatorname{const}(f^d)t^d$$

and similarly for $g$ denote their classical periods. Two Laurent polynomials are mutation equivalent if a composition of algebraic mutations relates them. The period conjecture. If

$$\pi_f(t)=\pi_g(t),$$

then $f$ and $g$ are mutation equivalent. This conjecture predicts that equality of classical periods accounts for the known equivalence of mirror Laurent polynomials; it is proved in the paper for normalized maximally mutable Laurent polynomials whose Newton polygons are $T$-polygons, but is not established in the stated generality.