[proof=RMCS vs. $n$] Seem to have $[Q]_\mathrm{res}$ that grows rapidly in cardinality in the side length. Here is sage code that plots these quivers:
Code used to find the restricted mutation classes (requires this version of sage):
Here is a table consisting of $n$ and the corresponding $[Q]_\mathrm{res}$ cardinalities.
| Side length | $|[Q]_\mathrm{res}|$ |
|---|---|
| $0$ | $1$ |
| $1$ | $1$ |
| $2$ | $2$ |
| $3$ | $19$ |
| $4$ | $17340$ |
For the case of a side length of $5$, the restricted mutation class appears to be extremely large (already $854{,}236$ quivers at depth $11$) yet none of the resultant quivers have edge multiplicity larger than $2$. These types of quivers may be good to check as an extreme case.
[/proof]