Joint distribution equivalence of mesh patterns S21 and S22

Let a mesh pattern be represented by a permutation together with a set of shaded cells, and let $\sim_{jd}$ denote joint distribution equivalence: two pairs of mesh patterns have the same joint occurrence distribution in permutations. The four mesh patterns in the claim are the increasing and decreasing length-three patterns with, respectively, the shaded-cell sets

$$\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2),(3,3)\}$$

and

$$\{(0,0),(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\}.$$

Joint distribution equivalence of S21 and S22. The first pair of mesh patterns, with points at $(1,1),(2,2),(3,3)$ and at $(1,3),(2,2),(3,1)$, satisfies $\sim_{jd}$, and the corresponding second pair, with the same point sets and the second displayed shading set, also satisfies $\sim_{jd}$.

These are additional pairs of mesh patterns with symmetric shadings suggested by computer evidence; the supplied text does not establish whether the asserted equivalences are proved or remain open.