The form conjecture for odd norm-perfect Eisenstein integers

Let $\alpha$ be an odd norm-perfect Eisenstein integer. An Eisenstein integer is odd when it is not divisible by $1-\omega$, and an odd prime is an odd Eisenstein integer that is prime. Two Eisenstein integers share no common non-unit factors when every common divisor is a unit.

Odd norm-perfect form conjecture. The integer $\alpha$ must have the form

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

where $\pi$ and $\gamma$ are both odd Eisenstein integers, $\pi$ is an odd prime, $k\equiv 2\pmod{3}$, and $\pi$ and $\gamma$ share no common non-unit factors.

The authors report that they have not found any odd norm-perfect Eisenstein integer and state this claim as the expected structure if one exists. Its status is unresolved in the source.