The prime divisibility proportion conjecture for sporadic Apéry-like sequences

Let $C(n)$ range over the Apéry-like sequences listed in Tables 2 and 3 of the source, including $(\eta)$ and the sequences $(\delta)$, $(\alpha)$, $(\epsilon)$, $(\zeta)$, and $(\gamma)$. A prime divides a sequence if it divides at least one term. The sporadic Apéry-like sequence proportion conjecture. If $C(n)$ is one of the sequences in Table 2 or is $(\eta)$, then the proportion of primes not dividing any $C(n)$ is

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

If $C(n)$ is one of $(\delta)$, $(\alpha)$, $(\epsilon)$, $(\zeta)$, or $(\gamma)$ from Table 3, then the proportion is

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

The conjecture summarizes heuristic and numerical observations concerning divisibility of sporadic Apéry-like sequences; no resolution is supplied in the source.