BIBD conjecture for binders of odd-prime equiangular tight frames

Let $p$ be an odd prime and let $(p,\ldots,p)$ be a tuple of length $s+1$. Let $\mathcal G(p,\ldots,p)$ denote the corresponding equiangular tight frame, and let its binder $\mathcal B(\mathcal G(p,\ldots,p))$ be the collection of simplices associated with that frame. A $(v,k,\lambda)$-BIBD is a balanced incomplete block design with $v$ points, block size $k$, and parameter $\lambda$.

Binder BIBD conjecture. The binder of $\mathcal G(p,\ldots,p)$ is a

$$\left(p^{2(s+1)},p^{s+1},\frac{p^{s+1}-1}{p-1}\right)\text{-BIBD}.$$

This conjecture generalizes the preceding results for the binder constructions, which establish the claimed design structure in lower-dimensional cases. The general assertion for every odd prime $p$ and every tuple length $s+1$ remains open.