Zauner's conjecture over finite fields

Let $d$ be a positive integer, let $q$ be a prime power, and let $\mathbb{F}_{q^2}$ be the quadratic extension of the finite field with $q$ elements. A unitary geometry on $\mathbb{F}_{q^2}$ is equipped with its standard sesquilinear form, and an equiangular tight frame (ETF) is a collection of vectors with equal-norm, equal-magnitude pairwise inner products that spans the space. Zauner's conjecture over finite fields. For every $d$, there exist infinitely many pairwise coprime $q$ such that a unitary geometry on $\mathbb{F}_{q^2}$ admits an ETF of $n=d^2$ vectors. This is posed as the finite-field analogue of Zauner's conjecture; it is an open problem for future investigation.