The graded-ring generalization of Kaplansky's conjectures

Let $G$ be a torsion-free group and let $R$ be a unital $G$-graded ring whose $G$-gradation is non-degenerate. If $R_e$ is a domain, then $G$ is said to satisfy the graded-ring generalization of Kaplansky's conjectures when the following assertions hold:

Graded-ring generalization of Kaplansky's conjectures. Every unit in $R$ is homogeneous, $R$ is a domain, and every idempotent in $R$ is trivial.

This conjecture extends the corresponding assertions for group rings to non-degenerate gradations. The paper establishes these assertions in several important cases, including unique product groups, commutative rings, and central elements; the general torsion-free case remains open.