The converse Hasse–Witt rank conjecture for curves in characteristic 2

Let $k$ be a perfect field of characteristic $2$, let $\Delta$ be a two-dimensional lattice polygon satisfying the conditions from Lemma~, and let $f$ be a $\Delta$-non-degenerate (or $\Delta$-toric) Laurent polynomial. Write $C'_f$ for the associated curve, $g$ for its genus, and $\rho$ for the quantity defined in the preceding results. For each solution $(i_0,j_0)\in\Delta\cap\mathbf{Z}^2$ of the system of congruences in, let $c_{i_0,j_0}$ denote the corresponding coefficient of $f$. Converse Hasse–Witt rank conjecture. The rank of the Hasse–Witt matrix of $C'_f$ is at least $g-\rho$, and this bound is attained if and only if $c_{i_0,j_0}=0$ for every such solution $(i_0,j_0)$. This conjectured converse would give a geometric interpretation of the stated condition on the coefficients, complementing the proved implication from the coefficient condition to the Hasse–Witt rank bound.