Asymptotic Mahler measure conjecture for Gaussian periods

Let $\alpha_n$ be the Gaussian-period algebraic integer defined in the paper, and let $\mathrm{m}(\alpha_n)$ denote its Mahler measure. For a property of odd integers $n$, say that it holds almost always if it fails for only $o(X)$ odd values of $n\leqslant X$ as $X\rightarrow +\infty$. Asymptotic Mahler measure conjecture. We have

$$\mathrm{m}(\alpha_n)\asymp n\log{\log{n}}$$

for all but $o(X)$ odd values of $n\leqslant X$ as $X\rightarrow +\infty$. This conjecture gives a precise typical growth rate for the Mahler measure of the Gaussian periods; the paper presents it as a conjectural refinement of the unconditional bounds $n\ll \mathrm{m}(\alpha_n)\ll n\log n$.