Bernstein's inequality for quantum differential operators at generic parameter

Let $q$ be a complex number that is not a root of unity, and let $G=\operatorname{GL}_2$ or $G=\operatorname{SL}_2$. A $\mathcal{D}_q(G)$-module is called strongly equivariant when it has the strong equivariance property used for quantum differential operators. Bernstein's inequality conjecture. The Gelfand–Kirillov dimension of any simple, strongly equivariant $\mathcal{D}_q(G)$-module is at least $\dim G$. The paper identifies this inequality as the main missing step toward proving the preceding non-root-of-unity extension conjecture; its status is therefore unresolved in the supplied text.