Conjecture on singular symmetric Bernoulli matrices with sharp exponential rate

Let $M_n^{\mathrm{sym}}$ be a random symmetric matrix whose upper-diagonal entries are iid Bernoulli variables, and let $p_n^{\mathrm{sym}}$ denote its probability of being singular. Sharp symmetric singularity conjecture.

$$p_n^{\mathrm{sym}}=(1/2+o(1))^n.$$

This is presented as a strengthening of the earlier conjecture that $p_n^{\mathrm{sym}}=o(1)$; the source gives no resolution for this sharp asymptotic.