The dominant-zero-column singularity conjecture for random combinatorial matrices

Let $M$ be a random combinatorial matrix, and let $\Omega_0$, $\Omega_r$, and $\Omega_c$ be respectively the events that $M$ has a zero column, contains two identical rows, and contains two identical columns.

Singularity-obstruction conjecture. If $c\in(0,1/2)$ is fixed and

$$2\log n\leq d\leq cn,$$

then

$$\mathbb{P}(M\text{ is singular})=(1+o(1))\mathbb{P}(\Omega_0).$$

The conjecture asserts that, in this sparse-to-intermediate regime, zero columns account asymptotically for all singularity. The supplied text gives no resolution, so the conjecture remains open.