The stubborn-matrix conjecture for random sign matrices

Call a $\{-1,1\}$ $n\times n$ matrix $M$ stubborn if every matrix obtained by switching any subset of its diagonal entries is nonsingular.

Stubborn-matrix conjecture. With probability $1-o(1)$, $M_n$ is stubborn.

This is a local-resilience question related to rank resilience; the source gives no resolution.