The singularity probability conjecture for random Bernoulli matrices

Let $A_n$ be an $n$ by $n$ random matrix with iid Bernoulli entries taking values $\pm 1$ with probability $1/2$, and let $p_n$ be the probability that $A_n$ is singular.

Singularity probability conjecture. The singularity probability satisfies

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

This conjecture asserts that the elementary lower bound arising from pairs of equal or opposite rows or columns is asymptotically best possible. At the time of the paper, the best known upper bound was exponential, namely $p_n=O((1/\sqrt{2})^n)$, so the conjectured asymptotic remained open.