The integral-spectrum probability conjecture for symmetric random sign matrices

Let $M_n^{\mathrm{sym}}$ be an $n\times n$ symmetric random sign matrix.

Integral-spectrum conjecture. The probability that all eigenvalues of $M_n^{\mathrm{sym}}$ are integers is

$$2^{-(0.5+o(1))n^2}.$$

Known results give substantially weaker upper bounds, so the conjecture remains open.