Conjecture on singularity of the Pell matrix at exceptional primes

Let $p=2n+1\ge7$ be an odd prime, so that $n=(p-1)/2$, and let $B_p(n-1)$ denote the matrix defined in the paper.

Singularity conjecture. The matrix $B_p(n-1)$ is singular if and only if

$$p\in\{13,31\}.$$

The conjecture is motivated by a computation through $10^6$ and the preceding equivalence between singularity and the congruence $Q_p\equiv2\pmod{p^2\mathbb{Z}}$. The supplied span gives no proof or disproof.