Vu's global rank resilience conjecture for Rademacher matrices

Let $M\in\{\pm1\}^{n\times m}$ with $m\geq n$, and let $\operatorname{Res}(M)$ be the least number of entry-flips needed to produce from $M$ a matrix whose rank is strictly less than $n$. Let $M\sim\mathcal{M}_n$, where $\mathcal{M}_n$ is the distribution of $n\times n$ Rademacher matrices, whose entries are independent random variables taking the values $-1$ and $1$ with probability $1/2$ each. Vu's global rank resilience conjecture. One has

$$\operatorname{Res}(M)=\left(1/2+o(1)\right)n$$

a.a.s. as $n\to\infty$. This strengthens the singularity estimate for random Rademacher matrices: exponential upper bounds on the singularity probability imply only a weaker lower bound of order $n/\log n$ for the resilience, whereas the conjecture predicts the optimal asymptotic value $n/2$.