The irreducible characteristic-polynomial conjecture for random sign matrices

Let $M_n$ be an $n\times n$ random matrix with independent Rademacher entries, and let its characteristic polynomial be $\det(tI-M_n)$.

Irreducibility conjecture. With probability $1-o(1)$, the characteristic polynomial of $M_n$ is irreducible.

The conjecture was independently associated with Vu, Wood, and Babai, and the source describes it as seemingly hard and unresolved.