Pattern-avoidance criterion for rational smoothness in types AI and AII

Let $AI$ and $AII$ denote the two types of symmetric varieties under consideration, and let an involution $\pi$ avoid all bad patterns in the sense of this section when the indices in each pattern are required to be permuted by the involution. In Schubert-variety pattern avoidance, that requirement is omitted.

Pattern-avoidance equivalence conjecture. In types $AI$ and $AII$, an involution $\pi$ avoids all bad patterns in the sense of this section if and only if it avoids all bad patterns in the sense used for Schubert varieties.

This would identify the bad-pattern criterion for the rational smoothness and related singularity properties of the $K$-orbit closures in these types with the corresponding criterion for Schubert varieties. The supplied text gives no evidence that this conjecture has been resolved.