The four-pattern involution equidistribution conjecture for length-2 mesh patterns

Let $p_1,p_2,p_3,p_4$ be the four length-$2$ mesh patterns displayed in the source, with mesh-cell sets respectively $\{(1,2),(1,1),(2,1),(0,0)\}$, $\{(2,2),(0,1),(1,1),(1,0)\}$, $\{(0,2),(1,1),(2,1),(1,0)\}$, and $\{(1,2),(0,1),(1,1),(2,0)\}$, and with points at $(1,1)$ and $(2,2)$. An involution is a permutation whose cycles have length at most $2$.

Four-pattern involution equidistribution conjecture. The patterns

$$\{p_1,p_2,p_3,p_4\}$$

are equidistributed on involutions.

The source notes that the first two patterns and the last two patterns are trivially equidistributed via reverse and complement. The conjectural content is the asserted equidistribution of the displayed collection on involutions.