Zhang's stratiform length conjecture for division algebras

Let $K$ be a field and let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension $d$. A chain

$$K=D_0\subseteq D_1\subseteq\cdots\subseteq D_m\subseteq Q(A)$$

of division subalgebras has stratiform length $m$ when each $D_i$ is infinite dimensional as a left $D_{i-1}$-vector space and finitely generated as a division algebra. Zhang's conjecture. Every such chain satisfies

$$m\leq d.$$

This conjecture proposes a noncommutative analogue of the fact that a field of transcendence degree $d$ cannot contain a chain of more than $d$ suitably independent finitely generated intermediate fields. It is motivated by the search for birational invariants of finitely generated domains of GK dimension two; the paper states that its methods do not prove the conjecture because they rely on two-sided estimates.