The exponent-one conjecture for 2-powerfully perfect numbers in the Gaussian and Eisenstein-type rings

Let $k$ be the exponent appearing in Theorems 2.2 and 2.4 for the classification of $2$-powerfully perfect numbers in the relevant quadratic rings. Exponent-one conjecture. The value of $k$ in Theorem 2.2 must be $1$. Similarly, if there is a $2$-powerfully perfect number in $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, then the value of $k$ in Theorem 2.4 must be $1$. The conjecture asserts a restriction on the exponent in the preceding classifications of $2$-powerfully perfect numbers; the supplied text gives no resolution.