The optimal singularity-probability conjecture for random symmetric Bernoulli matrices

Let $M_n$ be a random $n\times n$ symmetric matrix whose upper-diagonal entries are independent Bernoulli random variables taking the values $1$ and $-1$ with probability $1/2$ each, and let $q_n=\mathbb{P}(M_n\text{ is singular})$. The event that the first and last rows are equal up to a sign gives the lower bound $q_n\geq(2+o(1))^{-n}$. Optimal singularity-probability conjecture. One has

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

This conjecture asserts that the elementary row-dependence construction gives the correct asymptotic order. Before this work, the best known upper bound was only $q_n=O(2^{-n^c})$ for some unspecified constant $c>0$, while the matching lower bound was known.