Non-Archimedean functional Zauner conjecture

Let $\mathbb{K}$ be a non-Archimedean field satisfying Equation (FU). For each $d\in\mathbb{N}$, consider $\mathbb{K}^d$ with any non-Archimedean norm, and let $(\mathbb{K}^d)^*$ denote its dual. For vectors $\tau_1,\dots,\tau_{d^2}\in\mathbb{K}^d$ and functionals $f_1,\dots,f_{d^2}\in(\mathbb{K}^d)^*$, define

$$S_{f,\tau}(x)=\sum_{j=1}^{d^2}f_j(x)\tau_j.$$

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

$$|f_j(\tau_k)f_k(\tau_j)|=|d|,\qquad \forall\,1\leq j,k\leq d^2,\ j\neq k,$$

while $\|f_j\|=1$ and $\|\tau_j\|=1$ for every $1\leq j\leq d^2$. This is the non-Archimedean functional analogue of the Zauner conjecture, obtained by taking $n=d^2$ in the preceding question; the supplied text does not indicate whether it is known or open.