Fishburn and classical permutation enumeration conjecture for 1243 and 3124

Let $F_n(\sigma_1,\ldots,\sigma_k)$ denote Fishburn permutations of length $n$ avoiding the listed patterns, and let $S_n(\tau_1,\ldots,\tau_k)$ denote ordinary permutations in $S_n$ avoiding the listed patterns.

Enumeration conjecture. For every $n\geq 0$,

$$|F_n(1243,3124)|=|S_n(231,4123)|.$$

The source reports verification for $n\leq 15$ and cites related generating-tree results for the classical class, but the equality itself remains conjectural.