The non-commutative extension of the matrix power-sum theorem

Let $d>1$ and let $R$ be a finite ring, without assuming that $R$ is commutative. Define

$$S_k^d(R):=\sum_{A\in\mathbb{M}_d(R)}A^k.$$

Non-commutative power-sum conjecture. The theorem for finite commutative rings remains true for non-commutative $R$: $S_k^d(R)=0$ unless $d=2$, $\operatorname{card}(R)\equiv2\pmod4$, $1<k\equiv-1,0,1\pmod6$, and the unique nonzero element $e\in R$ satisfying $2e=0$ is idempotent; in the exceptional case,

$$S_k^d(R)=\begin{pmatrix}e&0\\0&e\end{pmatrix}.$$

The paper presents this as a final conjecture extending its commutative-ring theorem to finite non-commutative rings.