Existence conjecture for a projective-geometry triple-array family

Let $q$ be a prime power. A triple array with parameters $(r,c,n)$ has $r=\frac{q^3-1}{q-1}$, $c=\frac{q^4-1}{q-1}$, and $n=\frac{(q^4-1)(q^3-1)}{(q^2-1)(q-1)}$.

Existence conjecture. For any prime power $q$, there is a $\left(\frac{q^3-1}{q-1},\frac{q^4-1}{q-1},\frac{(q^4-1)(q^3-1)}{(q^2-1)(q-1)}\right)$-triple array.

This conjecture asserts existence for the parameter family associated with the paper's projective-geometry construction. The supplied text gives no resolution or status evidence beyond the conjectural formulation.