Hamada–Sachar conjecture on the p-rank of projective planes

Let $\Pi$ be a projective plane of order $p^s$, where $p$ is prime, and let its incidence matrix have rank over $\mathbb{F}_p$ called its $p$-rank. Hamada–Sachar conjecture. The $p$-rank of $\Pi$ is at least

$$\binom{p+1}{2}^{s}+1,$$

and equality holds if and only if $\Pi$ is Desarguesian. If true, this would establish that the projective-plane code discussed in the source has minimum block length and maximum rate among the specified binary storage codes; the source states that the conjecture remains unproved.