The local density conjecture for G-irregular primes

Given coprime positive integers $a$ and $d$, let ${\mathcal P}_G(d,a)$ be the set of G-irregular primes congruent to $a$ modulo $d$, and let $\pi(x;d,a)$ count primes $p\le x$ with $p\equiv a\pmod d$. Let $\delta(d,a)$ be the quantity defined in the paper's Proposition 1. The local G-irregular-prime conjecture. Asymptotically,

$${\mathcal P}_G(d,a)(x)\sim \left(1-\frac{\delta(d,a)}{\sqrt e}\right)\pi(x;d,a).$$

Numerical evidence is reported, and the conjecture implies the global density conjecture when $a=d=1$; it remains open.