The penult token-count interval conjecture for square Tak boards

Let $n\geq 6$. Write $M_n$ for the value given by Proposition, and let $x$ satisfy

$$M_n\leq x\leq n^2-4(n-2)-2.$$

A penult is a position on an $n\times n$ board with the stated token count from which the relevant game-theoretic condition holds. Penult token-count interval conjecture. For every such $n$ and $x$, there is a penult on an $n\times n$ board with $x$ tokens. The conjecture proposes that the token counts between the snake-diagram lower bound and the stated upper bound are all realized; the supplied text gives no resolution.