Bsila–Cox–Hugo–Styron–Zhuang fixed–pixed points equidistribution conjecture

Let $\mathcal P$ be the family

$$\mathcal P= \left\{ \begin{array}{ccc} \{132,312\}, & \{132,321\}, & \{213,231\},\\[2mm] \{123,132,312\}, & \{123,213,231\}, & \{123,312,321\},\\[2mm] \{132,312,321\}, & \{213,231,312\}, & \{213,231,321\} \end{array} \right\}.$$

For $\Pi\in\mathcal P$, let $\mathfrak S_n(\Pi)$ denote the permutations of length $n$ avoiding every pattern in $\Pi$, and let $\operatorname{fix}$ and $\operatorname{pix}$ be the fixed-point and pixed-point statistics, respectively. Bsila–Cox–Hugo–Styron–Zhuang conjecture. For all $n\geq 0$ and every $\Pi\in\mathcal P$, the statistics $\operatorname{fix}$ and $\operatorname{pix}$ are equidistributed over $\mathfrak S_n(\Pi)$. Equivalently,

$$\sum_{\pi\in\mathfrak S_n(\Pi)} x^{\operatorname{fix}(\pi)}=\sum_{\pi\in\mathfrak S_n(\Pi)} x^{\operatorname{pix}(\pi)}.$$

The conjecture concerns a common refinement of enumeration for restricted permutations and desarrangements. The source notes that Zhuang and his students independently obtained a bijective proof of the original conjecture, so the status of this formulation should be checked against that result.