The irreducibility conjecture for Stern polynomials with prescribed prime divisors

Let $B_n(t)$ denote the Stern polynomial indexed by the positive integer $n$. For a positive integer $n$, say that it has exactly $k$ prime divisors, with the source's convention for counting prime divisors. The prescribed-divisor irreducibility conjecture. For each $k\in\mathbb{N}_{+}$ there exists an integer $n$ with exactly $k$ prime divisors such that

$$B_n(t)\text{ is irreducible}.$$

This extends the prime-index irreducibility question to indices with any prescribed number of prime divisors. The supplied text gives no proof or resolution.