Pylyavskyy–first author conjecture on maximal connected skew shapes

Let $N$ be a positive integer, let $\mathcal{P}_{N}$ be the poset of skew shapes of size $N$ under Schur-positivity, and restrict to connected skew shapes. Two skew shapes are identified when they are equivalent in $\mathcal{P}_{N}$. For a skew shape $R$, let $R^\circ$ denote its antipodal rotation by $180^\circ$. Pylyavskyy–first author conjecture. The connected subposet of $\mathcal{P}_{N}$ has exactly $N$ maximal elements. More specifically, for each $l=1,\ldots,N$, there is a unique maximal element with $l$ rows; each maximal element is an equivalence class consisting of a single skew shape $R$, together with $R^\circ$ if $R\ne R^\circ$. If $[R]$ denotes the unique maximal element with $l$ rows, then, up to antipodal rotation, $R$ is obtained by taking the boxes whose interior or top-left corner is intercepted by a line $L$ from the bottom-left to the top-right corner of an $l$-box-high and $(N-l+1)$-box-wide grid. This conjecture gives the proposed answer to the still-open problem of determining the maximal connected elements of $\mathcal{P}_{N}$; the antipodal-rotation identification is a known equivalence, but the asserted maximality and uniqueness remain open.