The Bernoulli matrix singularity probability conjecture

Let $n$ be a large integer, and let $M_n$ be a random $n$ by $n$ matrix whose entries are independent Bernoulli random variables, each equal to $1$ or $-1$ with probability $1/2$. Define

$$P_n:= \mathop{\mathrm{P}}(\det(M_n)=0).$$

It is conjectured that Bernoulli matrix singularity conjecture.

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

The lower bound comes from the event that two rows or two columns are equal up to sign. The conjecture asserts that these elementary dependencies account asymptotically for the entire singularity probability; the paper proves only the upper bound $(3/4+o(1))^n$.