Point-count conjecture for the double octic O_5

Let $O_5$ be the double cover of $P^3$ branched over the octic displayed in the source. For each prime $p>2$, let $a_p$ denote the eigenvalue for $T_p$ of the newform of weight $4$ and level $5$, and let $\left(\frac{-1}{p}\right)$ be the Legendre symbol. Point-count conjecture for $O_5$. For all primes $p>2$, the threefold $O_5$ has

$$p^3+5p^2-(9+\left(\frac{-1}{p}\right))p+1-a_p$$

points over $F_p$. The conjecture was checked for $p<200$; the paper suggests proving it via a crepant resolution that is a rigid CalabiYau threefold, and relates it to the Tate conjecture.