Conjecture on the cut locus of lifted Fermat hypersurfaces

Let $\pi:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be the quotient map, let

$$X(d)=\left\{[z_0:z_1:\cdots:z_n]\in\mathbb{CP}^n:\sum_{i=0}^n z_i^d=0\right\},$$

and set $\tilde{X}(d)=\pi^{-1}(X(d))$. Here $\mathbb{Z}_d^{\star(n+1)}$ denotes the $(n+1)$-fold topological join of $\mathbb{Z}_d$, and $\times_{\mathbb{Z}_d}$ denotes the diagonal action of $\mathbb{Z}_d$. Cut-locus conjecture. The cut locus of $\tilde{X}(d)\subseteq\mathbb{S}^{2n+1}$ is

$$\mathbb{Z}_d^{\star(n+1)}\times_{\mathbb{Z}_d}\mathbb{S}^1.$$

This conjecture gives a geometric description of the cut locus of the lift of a Fermat-type complex hypersurface under the Hopf quotient. The supplied text does not state any partial results or resolution, so its status remains open.