Kahn–Komlós–Szemerédi conjecture on singularity of random Bernoulli matrices

Let $A_n$ be an $n\times n$ matrix with independent and identically distributed Rademacher random variables, taking values $\pm1$ with equal probability. Write

$$\gp_n=\mathbb{P}(A_n\text{ is non-invertible}).$$

The notation $a_n=o(b_n)$ means $\lim_{n\to\infty}a_n/b_n=0$.

Kahn–Komlós–Szemerédi conjecture.

$$\gp_n=\left(\frac12+o(1)\right)^n.$$

The conjectured leading contribution comes from pairs of identical rows or columns up to reversal of sign, and the statement concerns the principal obstruction to invertibility for random Rademacher matrices. The paper presents it as a well-known conjecture; its resolution status is not specified in the supplied text.