The pseudo-symmetry or F-equivalence conjecture for nonsingular toric Fano varieties

Let $T_N\operatorname{emb}(\Sigma)$ be a nonsingular toric Fano $d$-fold, with $N_{\mathbf R}=N\otimes_{\mathbf Z}\mathbf R$. A $d$-dimensional strongly convex rational polyhedral cone is a member of $\Sigma(d)$. The variety is pseudo-symmetric if there exist $d$-dimensional strongly convex rational polyhedral cones $\sigma,\sigma'\in\Sigma(d)$ such that

$$\sigma=-\sigma':=\{-x\in N_{\mathbf R}\mid x\in\sigma'\}.$$

Pseudo-symmetry or F-equivalence conjecture. Any nonsingular toric Fano $d$-fold is either pseudo-symmetric or F-equivalent to the $d$-dimensional projective space $\mathbf P^{d}$.

This conjecture would provide representatives for the classification of nonsingular toric Fano varieties under F-equivalence, which is generated by equivariant blow-ups and blow-downs through toric Fano varieties. The source states that it holds for $d\leq 4$ as a consequence of the known classification; no resolution is supplied for general $d$.