Resilience conjecture for nonsingularity of Bernoulli matrices

For a $\{-1,1\}$ matrix $M$, let $\operatorname{Res}(M)$ be the minimum number of entries that must be switched from $1$ to $-1$ or vice versa in order to make $M$ singular. Let $M_n$ be a random $n\times n$ Bernoulli matrix. Resilience conjecture. Almost surely,

$$\operatorname{Res}(M_n)=(1/2+o(1))n.$$

The conjecture predicts that making two rows equal is asymptotically the cheapest way to destroy nonsingularity. The source gives no resolution.