Fishburn-to-classical pattern enumeration conjecture for 2143 and 3124

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

Fishburn–classical enumeration conjecture. For every $n\geq 1$,

$$|F_n(2143,3124)|=|S_n(231,4123)|.$$

The source presents this as an open conjectured coincidence between Fishburn and ordinary pattern-avoiding permutation classes.