Random matrix prediction for regular primes in totally real abelian extensions

Let $F/\mathbb{Q}$ be a totally real abelian extension of order $m$, and suppose

$$\operatorname{Gal}(F/\mathbb{Q})\cong\bigoplus_{i=1}^n\mathbb{Z}/m_i\mathbb{Z}.$$

For primes $p\gg 0$ satisfying $p\nmid m\,\operatorname{disc}_F$, write $F$-regular for the condition that $p$ is $\chi$-regular for every character of $\operatorname{Gal}(F/\mathbb{Q})$. Regular-prime proportion conjecture. One estimates that

$$\operatorname{Prob}(p\text{ is }F\text{-regular})\approx \exp\left(-\frac{\prod_{i=1}^n\gcd(m_i,p-1)}{2}\right).$$

This extends the character-by-character regularity heuristic to totally real abelian extensions and is presented as a prediction for sufficiently large unramified primes.