Equidistribution of the pix and fix statistics on pattern-avoiding permutations

Let $Pi\subseteq\mathfrak{S}_{3}$ with $1\leq|\Pi|\leq3$. For each $n\geq0$, let $\mathfrak{S}_{n}(\Pi)$ denote the permutations avoiding every pattern in $\Pi$, and let $\operatorname{pix}$ and $\operatorname{fix}$ be the statistics on this avoidance class described in the paper.

Equidistribution conjecture. The statistics $\operatorname{pix}$ and $\operatorname{fix}$ are equidistributed over $\mathfrak{S}_{n}(\Pi)$ for all $n\geq0$ if and only if $\Pi$ is one of

$$\{132,312\},\ \{132,321\},\ \{213,231\},\ \{123,132,312\},\ \{123,213,231\},\ \{123,312,321\},\ \{132,312,321\},\ \{213,231,312\},\ \text{or }\{213,231,321\}.$$

This conjecture refines the paper's results on pattern-avoiding derangements. Computational evidence supports the stated classification, but the authors leave its resolution open; the exceptional set from the related theorem is $\Pi=\{132\}$, which is excluded from the listed cases.