Handelman's conjecture that every *-regular ring is unit-regular

A ring is $^*$-regular when it is von Neumann regular and its involution is proper; it is unit-regular when for every element $a$ there is a unit $u$ such that $a=aua$. Handelman's conjecture. Every $^*$-regular ring is unit-regular. The paper gives a positive answer for the family of Leavitt path algebras of arbitrary graphs, while the general ring-theoretic conjecture is presented as the problem under consideration.