The local obstruction conjecture for Bernoulli matrix singularity

Let $B_n(p)$ be an $n\times n$ random matrix with independent and identically distributed Bernoulli entries of mean $p$, and let $q_n(p)=\mathbb{P}[B_n(p)\text{ is singular}]$. Local obstruction conjecture. For fixed $p\in(0,1)$,

$$q_n(p)=(1+o_n(1))\mathbb{P}[B_n(p)\emph{ has two equal rows, two equal columns, a zero row, or a zero column}].$$

The conjecture asserts that singularity is asymptotically caused by these elementary local obstructions. The paper presents it as a natural extension of a folklore conjecture and notes that the available result of Tikhomirov does not determine the required sharp error term.