Fiorilli's generalized Skewes-number growth conjecture for quadratic residue races

For an integer $q$, let $\varrho(q)$ denote the number of square roots of $1$ modulo $q$, let $\operatorname{rad}(q)$ be its radical, and let $x_{q;R,NR}$ be the generalized Skewes' number for the race between quadratic residues and nonresidues modulo $q$. If $(q_n)_{n\geqslant 1}$ is a sequence of integers such that

$$\frac{\varrho(q_n)}{\log \operatorname{rad}(q_n)}\to\infty,$$

then Fiorilli's conjecture.

$$\log\log x_{q_n;R,NR}\asymp \frac{\varrho(q_n)}{\log \operatorname{rad}(q_n)}.$$

This predicts the growth of the first occurrence of the rare event in which quadratic residues collectively outperform nonresidues in a prime number race. The conjecture is attributed to Fiorilli; no resolution is given here.