Four-pattern Fishburn enumeration conjecture

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

Four-pattern enumeration conjecture. For every $n\geq 1$,

$$|F_n(1324,2143,1423,3124)|=\frac{(n+2)(n^2-2n+3)}{6}.$$

The source includes this among conjectured enumerations verified only for initial values, and gives no general resolution.