Rank conjecture for random sign matrices

Let $B_n$ be an $n\times n$ random sign matrix with independent entries taking the values $1$ and $-1$ with probability $1/2$ each, and let $k$ be any positive integer. Rank conjecture for random sign matrices.

$$\mathbb{P}\{\operatorname{corank}(B_n)\geq k\}=\mathbb{P}\{s_{n-k+1}(B_n)=0\}=\left(\frac12+o(1)\right)^{kn}.$$

This generalizes the preceding singularity conjecture from corank at least one to arbitrary fixed positive corank. The source presents it as a conjecture about the rank of random discrete matrices and gives no resolution for this statement.