The singularity probability conjecture for random symmetric Bernoulli matrices

Let $M_n$ be a random symmetric $n$ by $n$ matrix whose upper-diagonal entries are iid Bernoulli random variables taking values $\pm1$ with probability $1/2$, and let $q_n$ be the probability that $M_n$ is singular.

Symmetric-matrix singularity conjecture. The singularity probability satisfies

$$q_n=\left(\frac{1}{2}+o(1)\right)^n.$$

This conjecture predicts exponential decay for the singularity probability of random symmetric Bernoulli matrices. The paper proves decay faster than every polynomial, while exponential decay of the conjectured scale remained open.