The folklore conjecture on algebraic quantizations of Feigin–Odesskii varieties

Let $X$ be a Calabi–Yau curve and let $ξ$ be a simple vector bundle on $X$. Let $N(ξ)$ be the coarse moduli scheme of the moduli stack of non-splitting complexes $O_X\to V$ such that $V/\mathcal{O}_X\cong ξ$; it is identified with the projective space

$$N(ξ)\cong \mathbb{P}\operatorname{Ext}^1(ξ,\mathcal{O}_X).$$

An algebraic quantization of $N(ξ)$ is a quantization in the sense intended in the source. Folklore conjecture. Algebraic quantizations of $N(ξ)$ are Artin–Schelter regular algebras. This conjecture concerns the expected regularity of quantizations of Feigin–Odesskii Poisson varieties; the source presents it as folklore and gives no resolution.