The symmetric spectrum simplicity asymptotic

Let $M_n^{\mathrm{sym}}$ be an $n\times n$ symmetric random sign matrix, and let $s_n$ be the probability that its spectrum is not simple.

Symmetric spectrum simplicity conjecture.

$$s_n=(4+o(1))^{-n}.$$

The source records only a stretched-exponential upper bound, so the sharp asymptotic remains open.