The Apéry-number prime divisibility proportion conjecture

Let $A(n)$ denote the Apéry numbers, and let a prime divide the sequence if it divides at least one term $A(n)$. The Apéry-number proportion conjecture. The proportion of primes that do not divide any Apéry number $A(n)$ is

$$e^{-1/2}.$$

This is based on a heuristic treating the values $A(n)$ modulo a prime as independent and uniformly random, together with the Lucas congruences and a palindromic congruence. The source presents numerical evidence but no proof.