Finiteness conjecture for zero terms in the smallest-representation sequence

Let $p_n$ denote the $n$-th prime. For each $n\geq1$, let $s(n)$ be the smallest integer $k\geq3$ such that $p_n+k-3$ is represented by $F_k=x_1\cdots x_k+x_1+\cdots+x_k$, and set $s(n)=0$ if no such $k$ exists. The sequence begins

$$0,0,0,0,0,0,0,3,4,3,0,0,4,0,3,0,3,3,0,4,3,3,4,3,4,0,3,5,3,4,3,\ldots.$$

Finiteness conjecture. The sequence $s(n)$ contains only finitely many zero terms. This asserts that $p_n+k-3$ is represented by $F_k$ for some $k\geq3$ for all but finitely many primes $p_n$.