The rank-one projection conjecture at the exceptional parameter

Let $d\geq 3$ be an integer, let $|x\rangle$ be a vector on the unit sphere in $\mathbf R^{d-1}$, and construct matrices $A_m$ using $V_m=2|x\rangle\langle x|-I_{d-1}$ for every $m$. Rank-one projection conjecture. For every $d\geq 3$, there exists such a vector $|x\rangle$ for which the set

$$\{A_mA_n\}$$

is linearly independent when $t=-\frac{1}{d-1}$. This concerns the exceptional parameter where the highly symmetric choice of projection vector produces a degeneracy.