Hamada's conjecture on p-ranks of geometric designs

Let $D$ be a design with the parameters of a geometric design $G$ in $\operatorname{PG}(n,q)$ or $\operatorname{AG}(n,q)$, where $q=p^m$. Let the $p$-rank denote the rank of the incidence matrix over $\mathbb{F}_p$. Hamada's conjecture. The $p$-rank of $D$ is at least the $p$-rank of $G$, with equality if and only if $D$ is isomorphic to $G$. This is a generalization of the preceding projective-plane rank conjecture to designs with parameters arising from finite projective or affine geometries; no resolution is supplied in the source.