Shellability conjecture for $P_1$ intervals avoiding two patterns

Let $P_1$ be the permutation class denoted by $P_1$ in the source, and let $\pi$ be a permutation in $P_1$. Write $[\textbf{1},\pi]$ for the interval from the one-element permutation to $\pi$ in the permutation pattern poset. Shellability conjecture. If $\pi$ avoids both patterns $456123$ and $356124$, then

$$[\textbf{1},\pi]$$

is shellable. This is the positive formulation of the preceding obstruction claim; the source gives no evidence that it has been resolved.