Gil–Weiner enumeration conjecture for three Fishburn pattern classes II

Let $F_n(\sigma_1,\ldots,\sigma_k)$ denote the set of Fishburn permutations of length $n$ avoiding each listed pattern, and let $|\cdot|$ denote cardinality.

Gil–Weiner conjecture. For every $n\geq 0$,

$$|F_n(3214)|=|F_n(4132)|=|F_n(4213)|.$$

The source places this among open enumerative conjectures attributed to Gil and Weiner; no general proof or disproof is given.