The rigidity criterion for maximally mutable Laurent polynomials

Let $\mathcal N$ be a lattice and let $f i\tslashin i \mathbb C[\mathcal N]$ be a Laurent polynomial such that $\operatorname{Newt}(f)$ is a Fano polytope and the constant term of $f$ is zero. Define

$$S_f=\{(w,F)\mid f\text{ is mutable with respect to }(w,F)\}.$$

For a set $S$ of pairs $(w,F)$, with $w i\tslashin \mathcal M$ primitive and $F i\tslashin \mathbb C[w^\perp\cap\mathcal N]$, define

$$L_P(S)=\left\{f\ni\tslashin\mathbb C[\mathcal N]\,\left|\,\operatorname{Newt}(f)=P,\text{ the constant term of }f\text{ is zero, and }f\text{ is mutable with respect to }(w,F)\text{ for all }(w,F)\ni\tslashin S\right.\right\}.$$

A rigidity criterion for maximally mutable Laurent polynomials asserts that $f$ is a rigid MMLP if and only if $L_{\operatorname{Newt}(f)}(S_f)=\{f\}$.

This gives a local algebraic criterion for rigidity in terms of the mutations available from $f$. The source presents it as a conjectural candidate but gives no resolution status.