Conjectured equivalence of eight specified length-2 mesh patterns

A mesh pattern of length $2$ is a pair $(\tau,R)$, with $\tau$ a permutation of length $2$ and $R\subseteq[0,2]\times[0,2]$ the set of shaded boxes. Two patterns are equivalent when their occurrence distributions agree for every permutation length. Equivalence conjecture. Any two patterns belonging to the union of the following two sets are equivalent: the first set consists of the four patterns represented by $(\tau,R)$ with shaded-box sets $\{(0,2),(2,2),(1,1),(2,1),(0,0)\}$, $\{(0,2),(2,2),(1,1),(0,0),(1,0)\}$, $\{(1,2),(2,2),(1,1),(0,0),(2,0)\}$, and $\{(2,2),(0,1),(1,1),(0,0),(2,0)\}$; the second set consists of the four patterns with shaded-box sets $\{(0,2),(2,2),(0,1),(1,1),(2,0)\}$, $\{(0,2),(1,2),(1,1),(0,0),(2,0)\}$, $\{(0,2),(1,1),(2,1),(0,0),(2,0)\}$, and $\{(0,2),(2,2),(1,1),(1,0),(2,0)\}$. The displayed patterns in the source specify their underlying permutations and point placements. This is one of three remaining conjectural cases in the paper's classification; the source states that it would be valuable to investigate these unresolved cases.