Grassmannian enumeration conjecture for Fishburn pattern classes

Let $F_n(\sigma_1,\ldots,\sigma_k)$ denote the set of Fishburn permutations of length $n$ avoiding each listed pattern. A Grassmannian permutation is a permutation with at most one descent.

Grassmannian enumeration conjecture. For every $n\geq 1$,

$$|F_n(1324,2143,1423)|=|F_n(1324,2143,3124)|=|F_n(1324,1423,3124)|=2^n-n.$$

The source notes that $2^n-n$ is also the number of Grassmannian permutations of length $n$, and reports verification for $n\leq 17$; the asserted equalities remain open.