The division-subalgebra finite-dimensionality conjecture

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of finite GK dimension. Let $Q(A)$ be its quotient division algebra, and let $D$ be a division subalgebra of $Q(A)$ that is isomorphic to $Q(A)$. Finite-dimensionality conjecture. The division algebra $Q(A)$ is finite dimensional as a left $D$-vector space.

This is presented as a theoretically easier conjecture toward the stratiform length conjecture. The concluding remarks state that the methods developed in the paper do not establish the preceding conjecture, and no resolution of this easier statement is given.