Generalized Bernstein–Schwarzman conjecture for weak weighted projective spaces

Let $\Gamma$ and $\Gamma_1$ be commensurable complex crystallographic groups acting on $\mathbb{C}^n$, with equal linear parts $\mathrm{d}\Gamma_1=\mathrm{d}\Gamma$. Assume that $\Gamma$ is irreducible and generated by reflections. A weak weighted projective space is a projective toric variety of dimension $n$ whose fan contains $n+1$ rays; a genuine weighted projective space is the special case in which the primitive ray vectors generate the whole lattice. Generalized Bernstein–Schwarzman conjecture. The quotient $\mathbb{C}^n/\Gamma_1$ is a weak weighted projective space, and it is a genuine weighted projective space if and only if $\Gamma_1$ is also generated by reflections. The claim generalizes the Bernstein–Schwarzman conjecture to commensurable groups with the same linear part; the source provides the formulation but no resolution status.