Non-Archimedean Zauner conjecture

Let $\mathbb{K}$ be a non-Archimedean field satisfying Equation (FU). For each $d\in\mathbb{N}$, consider vectors $\tau_1,\dots,\tau_{d^2}\in\mathbb{K}^d$ and the operator

$$S_\tau:\mathbb{K}^d\ni x\mapsto\sum_{j=1}^{d^2}\langle x,\tau_j\rangle\tau_j\in\mathbb{K}^d.$$

Non-Archimedean Zauner conjecture. For every $d\in\mathbb{N}$, there exist such vectors satisfying $\langle\tau_j,\tau_j\rangle=1$ for all $1\leq j\leq d^2$, with $S_\tau$ diagonalizable, and

$$|\langle\tau_j,\tau_k\rangle|^2=|n|,\qquad\forall\,1\leq j,k\leq d^2,\ j\neq k.$$

This is presented as the non-Archimedean analogue of the Zauner conjecture for equiangular configurations in Hilbert spaces and related settings. The preceding question asks more generally for which pairs $(d,n)$ configurations satisfying the non-Archimedean Welch-bound conditions exist; the conjecture specializes to $n=d^2$.