The singularity probability conjecture for random sign matrices

Let $A$ be chosen uniformly from the set of $n\times n$ matrices with entries in $\{-1,+1\}$. Equivalently, $A\in\{-1,+1\}^{[n]^2}$, and singularity means $\det(A)=0$ in $\mathbb{Z}$. Singularity probability conjecture. As $n\to\infty$, the proportion of singular matrices satisfies

$$\frac{\left\lvert\{A\in\{-1,+1\}^{[n]^2}:\det(A)=0\text{ in }\mathbb{Z}\}\right\rvert}{2^{n^2}}\sim\left(\frac12+o(1)\right)^n.$$

This conjecture predicts the asymptotic decay of the singularity probability for uniformly random sign matrices. The source describes it as an old, plausible, and still unproved conjecture; determining the correct order of decay remains open.