Fishburn and classical permutation enumeration conjecture for 1243 and 2134

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 $n\geq 0$,

$$|F_n(1243,2134)|=|S_n(123,3241)|.$$

The source reports verification for $n\leq 15$ and notes related ascent-sequence interpretations of the right-hand class, but no proof of the conjectured equality.