Asymptotic comparison conjecture for Gaussian periods and cyclovarieties

For odd integers $n$, let $\alpha_n$ be the Gaussian-period algebraic integer and let $C_{\kappa(n)}$ be the associated cyclovariety, with Mahler measures denoted by $\mathrm{m}(\alpha_n)$ and $\mathrm{m}(C_{\kappa(n)})$. A property of odd integers holds almost always if it fails for only $o(X)$ odd values of $n\leqslant X$ as $X\rightarrow +\infty$. Asymptotic comparison conjecture. For odd $n$,

$$\frac{\mathrm{m}(\alpha_n)}{n}\asymp \mathrm{m}(C_{\kappa(n)})$$

almost always. Together with the conjectured lower bound for $\mathrm{m}(C_{\kappa(n)})$, this is intended to imply the paper's main conjecture on the typical growth of $\mathrm{m}(\alpha_n)$; no resolution is supplied in the given text.