Experimental coincidence classification conjecture for mesh patterns

Let $n$ be a nonnegative integer, and consider mesh patterns of size $n$. Experimental classification tests coincidence using permutations of a specified size.

Experimental coincidence classification conjecture. The mesh patterns of size $n$ can be coincidence classified by experimental classification with permutations of size

$$(n + 1)^2 + n.$$

This conjecture extends the experimentally verified cases for mesh patterns of sizes $0$, $1$, $2$, and $3$, which were classified using permutations of sizes $1$, $3$, $5$, and $10$, respectively. It proposes a general bound for obtaining coincidence classifications experimentally, but the source provides no proof or resolution.