Parker, Rushall and Hunt's form conjecture for odd norm-perfect Eisenstein integers

An odd norm-perfect Eisenstein integer is an odd Eisenstein integer whose norm is perfect. It is primitive if it has no proper norm-perfect Eisenstein-integer divisor up to association.

Parker, Rushall and Hunt's conjecture. Every odd norm-perfect Eisenstein integer has the form

$$\alpha=\psi^k\gamma^3,$$

where $\psi$ is an odd Eisenstein prime, $k$ is a rational integer satisfying $k\equiv 2\pmod{3\mathbb{Z}}$, and $\gamma$ is an odd Eisenstein integer coprime to $\psi$.

This conjecture proposes the analogue of the Euclid–Euler structural description for odd norm-perfect Eisenstein integers. The supplied text gives no evidence that it has been proved or disproved.