McNamara–Pylyavskyy maximal-elements conjecture for connected skew shapes

Let $\mathcal{P}_N$ be the poset of skew shapes with $N$ cells, ordered by Schur-positivity, and let $P_{R,S}$ be the box diagonal diagram with $R$ rows, $S$ columns, and $N=R+S-1$ cells, formed from the cells whose interior or top-left corner is intercepted by the line from the bottom-left to the top-right corner of an $R$-by-$S$ grid. McNamara–Pylyavskyy's maximal-elements conjecture. In the subposet of $\mathcal{P}_N$ consisting of connected skew shapes, there are exactly $N$ maximal elements, namely

$$P_{R,N-R+1}\quad\text{for }R=1,\ldots,N.$$

This conjecture identifies the proposed Schur-maximal connected skew shapes; the paper uses it to motivate the study of equitable ribbons and proves related special cases, but does not establish the conjecture in full generality.