Conjectured joint equidistributions for pairs X^(4)_7 and Y^(4)_7

Let a mesh pattern be specified by a set of marked points in a $3\times3$ grid together with a set of shaded cells; let $\sim_{jd}$ denote equality of joint distributions for the corresponding mesh-pattern statistics. The conjecture for pairs $X^{(4)}_7$ and $Y^{(4)}_7$. The two displayed pairs of mesh patterns are jointly equidistributed:

$$\operatorname{pattern}(1/1,2/2,3/3;0/0,0/1,3/0,2/0,2/1,2/2,2/3,3/1,3/3)\sim_{jd}\operatorname{pattern}(1/1,2/3,3/2;0/0,0/1,3/0,2/0,2/1,2/2,2/3,3/1,3/3),$$

and

$$\operatorname{pattern}(1/1,2/2,3/3;0/0,0/2,0/3,1/0,1/2,1/3,2/2,3/2,3/3)\sim_{jd}\operatorname{pattern}(1/1,2/3,3/2;0/0,0/2,0/3,1/0,1/2,1/3,2/2,3/2,3/3).$$

These are conjectured joint equidistributions of mesh patterns; the source provides no resolution, so their status remains open.