The prime-index irreducibility conjecture for Stern polynomials

Let $B_p(t)$ be the Stern polynomial indexed by a prime number $p$. The prime-index irreducibility conjecture. For every prime number $p$, the polynomial $B_p(t)$ is irreducible.

The conjecture was checked computationally for the first million primes, but the supplied text gives no general proof or resolution.