Conjecture on the spark and bender of Paley equiangular tight frames

Let $\Phi_q$ be a Paley ETF with $q=p^s$.

Paley ETF spark and bender conjecture. The spark of $\Phi_q$ is $((p+1)/2)^s$ and the bender is a $(p^s, ((p+1)/2)^s,\lambda)$-BIBD.

This conjecture predicts the spark and associated block design parameters for Paley ETFs, including those with non-cyclic additive groups. It is motivated by one infinite class and one example outside that class; the source provides no resolution, so the conjecture remains open.