Diagonal-switching conjecture for Bernoulli matrices

Let $M$ be a $\{-1,1\}$ matrix. Call $M$ good if every matrix obtained by independently switching any subset of its diagonal entries is nonsingular. Let $M_n$ be a random $n\times n$ Bernoulli matrix. Diagonal-switching conjecture. Almost surely, $M_n$ is good. This is a local-resilience question for nonsingularity: it asks whether changing arbitrary diagonal signs cannot create a singular matrix. The supplied source does not report a resolution.