Unit regularity conjecture for proatomic $*$-regular rings

Let $R$ be a $*$-regular ring, and let $\overline{L}(R)$ denote its associated modular ortholattice. Suppose that $\overline{L}(R)$ is proatomic, meaning that it has the proatomic property considered in the source, and has an orthogonal large partial $n$-frame. A ring is unit regular if for every $a\in R$ there is a unit $u$ such that $aua=a$; it is directly finite if $xy=1$ implies $yx=1$.

Unit-regularity conjecture. If $R$ is $*$-regular and $\overline{L}(R)$ is proatomic with an orthogonal large partial $n$-frame, then $R$ is unit regular. Every representable ring is directly finite, and is unit regular if it is simple.

The source presents these assertions together in the discussion of regularity properties of representable rings. It gives no resolution, although it attributes a related direct-finiteness question to Handelman.