Hamada's p-rank conjecture for designs with Singer parameters

Let ${\mathcal D}$ be a symmetric design with Singer parameters

$$\left(\frac{q^d-1}{q-1},\frac{q^{d-1}-1}{q-1},\frac{q^{d-2}-1}{q-1}\right),$$

where $q=p^s$ and $p$ is prime. Write ${\rm rank}_p{\mathcal D}$ for the dimension over ${\mathbb F}_p$ of the code of the design. Hamada's conjecture. One has

$${\rm rank}_p{\mathcal D}\geq {\binom{p+d-2}{d-1}}^s+1,$$

with equality if and only if ${\mathcal D}$ is the development of a classical Singer difference set. This conjecture gives a lower bound for the $p$-rank of such designs and characterizes the equality case; the source states that it remains open.