The strong form of the p-adic Zauner conjecture

Let $p$ be a prime, let $d\in\mathbb{N}$, and let $\langle\cdot,\cdot\rangle$ and $\|\cdot\|$ denote the inner product and norm on $\mathbb{Q}_p^d$. The vectors $\tau_1,\ldots,\tau_{d^2}$ lie in $\mathbb{Q}_p^d$, and $|\cdot|$ is the $p$-adic absolute value. The strong form of the p-adic Zauner conjecture. For every $d\in\mathbb{N}$, there exist vectors $\tau_1,\ldots,\tau_{d^2}\in\mathbb{Q}_p^d$ such that $\langle\tau_j,\tau_j\rangle=1$ for all $1\leq j\leq d^2$, there is a $b\in\mathbb{Q}_p$ satisfying

$$\sum_{j=1}^{d^2}\langle x,\tau_j\rangle\tau_j=bx,\qquad\forall x\in\mathbb{Q}_p^d,$$

$|\langle\tau_j,\tau_k\rangle|^2=|n|$ for all $1\leq j,k\leq d^2$ with $j\neq k$, and $\|\tau_j\|=1$ for all $1\leq j\leq d^2$. This strong form additionally requires all vectors to have norm one. The supplied text does not state whether it has been proved or disproved.