CN-hyperenergeticity conjecture for finite CC-rings

Let $R$ be a finite CC-ring, meaning that the centralizer of every non-central element of $R$ is commutative, and let $\Gamma_R$ be its commuting graph. A graph is CN-hyperenergetic if its CN-energy exceeds that of the complete graph on the same number of vertices. CN-hyperenergeticity conjecture for finite CC-rings. If $R$ is a finite CC-ring, then $\Gamma_R$ is not CN-hyperenergetic. This extends a conjecture attributed in the source to fn24 and concerns common-neighborhood energy for commuting graphs of finite rings; its resolution is not specified here.