Smoothness conjecture for Hessenberg Schubert varieties

Let $h\colon [n]\to[n]$ be a Hessenberg function, let $\Omega_{w,h}$ be the Hessenberg Schubert variety indexed by a permutation $w$, and let $\Gamma_{w,h}$ be the subgraph of the GKM graph $\Gamma_h$ induced by the torus fixed points in $\Omega_{w,h}^T$. For a pattern $\overline{p}$, say that $w$ avoids $\overline{p}$ if it does not contain that associated pattern.

Smoothness conjecture. The following statements are equivalent: $\Omega_{w,h}$ is smooth; $\Gamma_{w,h}$ is a regular graph; and $w$ avoids all the patterns $\overline{2143}$, $\overline{1324}$, $\overline{1243}$, $\overline{2134}$, $\overline{1423}$, $\overline{2314}$, $\overline{25314}$, $\overline{24315}$, $\overline{14325}$, and $\overline{15324}$.

The paper proves that regularity of $\Gamma_{w,h}$ and avoidance of the listed patterns are equivalent, and that both are necessary for smoothness. The conjecture asserts that this necessary pattern-avoidance condition is also sufficient, completing the proposed combinatorial characterization of smoothness.