The graded-equivalence conjecture for isomorphic quantum projective spaces

Let $\mathbb{P}^{n-1}_{\omega}$ and $\mathbb{P}^{n-1}_{\omega'}$ be quantum projective spaces with associated graded algebras $S_{\omega}$ and $S_{\omega'}$, respectively. Here $Gr(S_{\omega})$ denotes the category of graded modules over $S_{\omega}$, and similarly for $S_{\omega'}$. Graded-equivalence conjecture. If

$$\mathbb{P}^{n-1}_{\omega}\cong\mathbb{P}^{n-1}_{\omega'},$$

then

$$Gr(S_{\omega})\cong Gr(S_{\omega'}).$$

The conjecture asks whether isomorphic quantum projective spaces necessarily have equivalent categories of graded modules. The paper explains that the analogous implication from birational equivalence, even with isomorphic point varieties, can fail, while the question for isomorphic quantum projective spaces remains open.