Class 1 conjecture for relative non-commuting graphs of proper non-commutative subrings

Let $R$ be a ring and let $S$ be a proper non-commutative subring of $R$. The relative non-commuting graph $\Gamma_{S,R}$ has chromatic index equal to its maximum vertex degree.

Class 1 conjecture. The relative non-commuting graph $\Gamma_{S,R}$ is of class $1$.

For a graph, being of class $1$ means that its chromatic index equals its maximum vertex degree, in contrast with class $2$, where the chromatic index is one larger. The paper establishes that the non-commuting graph of a ring is of class $2$; the asserted class $1$ property for proper relative non-commuting graphs is presented as a conjecture.