Unique smooth minimum-degree toric surface of maximal Picard rank in an m×m box

Fix $m\in\mathbb{Z}_{>0}$. Consider toric surfaces $S$ with ample anticanonical divisor $-K_S$, without imposing conditions on their singularities, such that the anticanonical polygon of monomials in $H^0(S,-K_S)$ lies in an $m\times m$ box. Among these surfaces, consider those with maximum Picard rank. The unique smooth minimum-degree surface conjecture. Among the surfaces of maximum Picard rank, there is a unique surface $S$ that is smooth and has minimum anticanonical degree. This statement is reported to hold for box sizes up to seven based on the classification; the conjecture proposes it for every positive integer $m$.