Artin's classification conjecture for function skewfields of noncommutative surfaces

Let $A$ be a noetherian, connected graded domain over ${\Bbbk}$ with $\operatorname{GKdim} A=3$. Its graded quotient ring has the form $Q_{gr}(A)\cong D[t,t^{-1};\alpha]$, where $D=D_{gr}(A)$ is the function skewfield of $A$. A noncommutative surface is regarded as the category $\operatorname{qgr-\!}A$. Artin's classification conjecture. The only function skewfields of noncommutative surfaces are: (i) division rings $D$ finite dimensional over their centres $F=Z(D)$, which are then fields of transcendence degree two; (ii) division rings of fractions of Ore extensions ${\Bbbk}(X)[z;\sigma,\delta]$ for some curve $X$; and (iii) the function skewfield $D=D_{gr}(S)$ of a Sklyanin algebra $S=S(a,b,c)$, assuming that $S$ is not a finite module over its centre. Artin further asked for a classification of the noncommutative surfaces $\operatorname{qgr-\!}A$ within each birational class, namely the connected graded noetherian algebras $A$ with a fixed division ring $D_{gr}(A)$ from this list. This conjecture proposes a birational classification of noncommutative surfaces and reduces their study to the classification of orders within the proposed function-skewfield classes; the source gives no resolution of the conjecture.