MSN-integrality conjecture for finite non-commutative CC-rings

Let $R$ be a finite non-commutative CC-ring, meaning that all centralizers of non-central elements of $R$ are commutative, and let $\Gamma_R$ be its commuting graph. Let MSN-integral mean that the graph's minimum second-neighborhood spectrum consists entirely of integers, and let MSN-hyperenergetic mean that its minimum second-neighborhood energy exceeds that of the complete graph of the same order. MSN-integrality conjecture for finite non-commutative CC-rings. If $R$ is a finite non-commutative CC-ring, then $\Gamma_R$ is MSN-integral but not MSN-hyperenergetic. The conjecture is motivated by the examples treated in Sections 2–4, which are reported to have both properties; its general resolution is not specified here.