The Gaussian-integral-spectrum conjecture for random sign matrices

Let $M_n$ be an $n\times n$ random matrix with independent Rademacher entries.

Gaussian-integral-spectrum conjecture. The probability that all eigenvalues of $M_n$ are Gaussian integers is

$$2^{-(1+o(1))n^2}.$$

The source presents this as the non-symmetric analogue of the integral-spectrum conjecture and gives no resolution.