Power-sum conjecture for matrix rings over finite commutative rings

Let $d>1$ and let $R$ be a finite commutative ring. For $k\geq 1$, write $S_k(\mathbb{M}_d(R))=\sum_{A\in\mathbb{M}_d(R)}A^k$. The unique element $e\in R\setminus\\{0\\}$ such that $2e=0$ is assumed to be defined when the stated exceptional conditions hold. Power-sum conjecture for matrix rings. $S_k(\mathbb{M}_d(R))=0$ unless $d=2$, $|R|\equiv 2\pmod{4}$, $1<k\equiv -1,0,1\pmod{6}$, and $e$ is idempotent. In that exceptional case,

$$S_k(\mathbb{M}_d(R))=\begin{pmatrix}e&0\\\\0&e\end{pmatrix}.$$

This conjecture extends the known calculation for matrix rings over finite fields and proposes a complete description of when the power sum can be nonzero for matrices over a finite commutative ring. The source provides no resolution, so its status remains open.