Commutative Local Rings over Which Every Upper-Triangular Matrix Is the Sum of an Idempotent and a q-Potent That Commute

We study commutative local rings over which every upper-triangular matrix is the sum of an idempotent and a $q$-potent that commute. For Galois rings and rings of the form $\mathbb{F}_{p^{k}}[x]/\langle x^{r} \rangle$, necessary and sufficient criterion are provided.