Bean–Tannock–Ulfarsson purity conjecture for boundary-grid downcore graphs

Let $\pi$ be a $123$-avoiding permutation, and let its boundary grid be the associated skew Young diagram. The downcore graph has the boxes of the boundary grid as vertices, with an edge between $(i,j)$ and $(k,\ell)$ when either $i<k$ and $j>\ell$, or $i>k$ and $j<\ell$, and both opposite-corner boxes $(i,\ell)$ and $(k,j)$ lie in the skew Young diagram. A graph is pure if every maximal independent set has the same size. Bean–Tannock–Ulfarsson's purity conjecture. The downcore graph of the boundary grid of $\pi$ is pure if and only if $\pi$ avoids $2143$. This conjecture concerns a structural characterization of $123$-avoiding permutations through their boundary-grid graphs; the supplied text gives no resolution, so its status remains open.