Pattern-avoidance image conjecture for looms

Given a set of permutations $S$, let $\operatorname{Av}(S)$ denote the set of permutations avoiding the patterns in $S$. Let $\mathfrak{L}_{n,m}$ be the family of looms and let $\gamma_{n,m}$ be the map from looms to permutations. Pattern-avoidance image conjecture. For every $n,m\geq 1$, there exists a set of permutations $S_{n,m}$ such that

$$\gamma_{n,m}(\mathfrak{L}_{n,m})=\operatorname{Av}(S_{n,m}).$$

This predicts that the image of the loom family is a permutation class described by pattern avoidance; the supplied text gives no evidence of resolution.