The asymptotic singularity conjecture for random 0/1-matrices

Let $P_d$ be the probability that a random $0/1$-matrix of size $(d+1)\times(d+1)$ is singular; equivalently, in the associated $\pm1$ model, let $P_d$ denote the probability that a random $\pm1$-matrix of size $(d+1)\times(d+1)$ is singular. Singularity-probability conjecture. The probability is asymptotically dominated by zero rows or columns and by pairs of equal rows or equal columns:

$$P_d\sim 2\binom{d+1}{2}\frac{1}{2^d}\sim\frac{d^2}{2^d}.$$

Equivalently, a singular random $\pm1$-matrix has, with probability tending to one in the stated sense, two rows or two columns that are equal or opposite. The conjecture refines the known lower and upper bounds, but the source gives no resolution.