Partition characterization of irreducible zero mutable Laurent polynomials

Let $\triangle$ be a lattice polygon, let $e$ run over its edges, and let $\ell(e)$ denote the lattice length of $e$. For a zero mutable Laurent polynomial $f$ with Newton polygon $\triangle$, let $\mathbf p_e(f)=(s_{e,l})_{l=1}^{w}$ be the decreasing partition associated with the divisibility steps along $e$.

Partition characterization conjecture. Irreducible zero mutable Laurent polynomials on $\triangle$ are in one-to-one correspondence with tuples $(\mathbf p_e(f))_e$ of decreasing partitions of $\ell(e)$ such that

$$\sum_e\sum_{k=1}^{w}\sum_{l=k}^{w}s_{e,l}$$

equals the number of lattice points of $\triangle$.

The preceding proposition shows that every zero mutable Laurent polynomial yields such partitions, and the proposition also says the polynomial is determined by them. The conjecture asks whether the stated numerical condition is sufficient; the source mentions evidence from the classification section but gives no general proof.