The rank-resilience conjecture for random sign matrices

For a square $\pm1$ matrix $M$, define $\operatorname{Res}(M)$ to be the minimum number of entries that must be switched from $1$ to $-1$ or vice versa to make $M$ singular.

Rank-resilience conjecture. With probability $1-o(1)$,

$$\operatorname{Res}(M_n)=\left(\frac12+o(1)\right)n.$$

The source notes a partial result but does not report a resolution of the conjecture.