Strict Bott-Samelson desingularization conjecture for Schubert varieties

Let $X^w$ be the Schubert variety indexed by $w\in S_n$, where $n>4$, and let $\Gamma_n$ be the recursively defined set of permutations to avoid, with $\Gamma_5=\\{45312\\}$ and

$$\Gamma_n=\\{\pi_n\\}\cup\Gamma^T_{n-1},$$

where $\Gamma^T_{n-1}$ consists of the translations of permutations in $\Gamma_{n-1}$ and $\pi_n=n324\cdots(n-3)(n-1)(n-2)1$.

Strict Bott-Samelson desingularization conjecture. If $w\ngeq v$ for every $v\in\Gamma_n$, then $X^w$ can be desingularized using a sequence of strict Bott-Samelson resolutions.

This conjecture proposes that the recursively defined forbidden permutations exactly capture the obstructions to desingularizing singular Schubert varieties by successive strict Bott-Samelson resolutions. The paper gives examples and motivating patterns in low rank, but the general assertion remains open.