Catalan-convolution conjecture for pairs of Fishburn pattern classes

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

$$C_n=\frac{1}{n+1}\binom{2n}{n}$$

be the $n$th Catalan number.

Catalan-convolution conjecture. For every $n\geq 1$,

$$|F_n(2413,2431)|=|F_n(2431,3241)|=\sum_{k=1}^n\binom{n-1}{k-1}C_{n-k}.$$

The source reports verification for $n\leq 14$ and relates the sequence to known Catalan-binomial-transform interpretations, but does not prove the conjecture.