Solitary powers conjecture in quadratic integer rings

Let $K$ be the set of parameters defining the quadratic integer rings under consideration, let $d\in K$, let $p$ be an integer prime, and let $k$ and $n$ be positive integers. An element is $n$-powerfully solitary if it has no distinct $n$-powerful friend in $\mathcal O_{\mathbb{Q}(\sqrt{d})}$. For a prime $\pi$ in $\mathcal O_{\mathbb{Q}(\sqrt{d})}$, write $\overline{\pi}$ for its conjugate.

Solitary powers conjecture. The element $p^k$ is $n$-powerfully solitary in $\mathcal O_{\mathbb{Q}(\sqrt{d})}$ for every positive integer $n$. More strongly, $\pi^{\alpha_1}\overline{\pi}^{\alpha_2}$ is necessarily $n$-powerfully solitary in $\mathcal O_{\mathbb{Q}(\sqrt{d})}$ whenever $\alpha_1,\alpha_2,n\in\mathbb{N}$.

Parts of this conjecture are resolved by part (e) of Theorem 2.2 and Theorem 3.2, but the statement is not resolved in full in the source.