The double-exponential generalized Skewes-number conjecture

Let $q$ be a modulus, let $q'$ be its radical, and let $\rho(q)$ denote the quantity used in the paper's construction of highly biased prime number races. Define the generalized Skewes number for that race by

$$x_q:=\inf\{x:(\rho(q)-1)\pi(x;q,NR)<\pi(x;q,R)\}.$$

Here $R$ and $NR$ denote the quadratic-residue and nonresidue competitors, respectively.

Double-exponential generalized Skewes-number conjecture. As $\rho(q)/\log q'$ tends to infinity,

$$\log\log x_q\asymp\frac{\rho(q)}{\log q'}.$$

This predicts the rate at which the generalized Skewes number grows for the highly biased races constructed in the paper. The source presents it as a speculation based on analogous arguments, and gives no resolution; its status is open.