Asymptotic conjecture for \ell-Genocchi irregular primes in arithmetic progressions

Let $\ell$ be an odd prime, let $1\leq a<d$ with $(a,d)=1$, and let $\mathcal P_G(d,a)(x)$ count the primes $p\leq x$ satisfying $p\equiv a\pmod d$ that are $\ell$-Genocchi irregular. Write $\delta_{d,a}$ for the explicit constant defined by the paper, and let $\varphi$ denote Euler's totient function. \ell-Genocchi arithmetic-progression conjecture. As $x$ tends to infinity,

$$\mathcal P_G(d,a)(x)\sim \left(1-\frac{\delta_{d,a}}{\sqrt e}\right)\frac{x}{\varphi(d)\log x}.$$

The paper proves a positive lower-density result in every primitive residue class, while this sharper asymptotic is motivated by numerical data and a Siegel-type heuristic.