The cubic-residue occurrence conjecture over prime-power rings

Let $p$ be prime, let $m$ be a positive integer, and write $\mathbb{Z}_{p^m}$ for the integers modulo $p^m$. Let $CR$ denote a cubic residue and let $[i]$ denote the $i$th class in the classification of cubic residues over $\mathbb{Z}_{p^m}$. Cubic-residue occurrence conjecture. If $p \equiv 2 \pmod 3$, the occurrence of $CR$s in class $[i]$ of $\mathbb{Z}_{p^m}$ is $p^{2(i-1)}$; if $p \equiv 1 \pmod 3$, it is $3p^{2(i-1)}$, where

$$1\leq i \leq \left\lfloor \frac{m-1}{3} \right\rfloor+1.$$

The claim is based on computational data and comparison with known counting formulas; no resolution status is supplied.