Singularity probability conjecture for random fixed-row-sum matrices

Let $n$ be even, and let $Q_n$ be a random $n$ by $n$ $(0,1)$ matrix whose rows are independent vectors with exactly $n/2$ zero components. Fixed-row-sum singularity probability conjecture. The probability that $Q_n$ is singular satisfies

$${\bf P}(Q_n\text{ is singular})=\left(\frac{1}{2}+o(1)\right)^n.$$

The theorem in the source proves only the bound $O(n^{-C})$ for every $C>0$, while equal rows give the displayed lower bound. Thus the conjecture predicts that this elementary obstruction has the correct asymptotic order.