The pattern-avoidance inequalities conjecture for restricted-cycle permutations

For each $n\geq 1$, let $a_n^{\{1,3\}}(\sigma)$ count length-$n$ permutations avoiding the pattern $\sigma$ whose cycle lengths belong to $\{1,3\}$, and let $a_n^{\{1,2,3\}}(\sigma)$ count length-$n$ permutations avoiding $\sigma$ whose cycle lengths belong to $\{1,2,3\}$. Pattern-avoidance inequalities conjecture. For $n\geq 1$, one has

$$a_n^{\{1,3\}}(132) \leq a_n^{\{1,3\}}(321)$$

and

$$a_n^{\{1,2,3\}}(123) \leq a_n^{\{1,2,3\}}(132)\leq a_n^{\{1,2,3\}}(321)\leq a_n^{\{1,2,3\}}(231).$$

These inequalities are conjectured from numerical evidence in the cited tables; the supplied text does not establish them or report a resolution.