Positive-density conjecture for \ell-Genocchi regular primes in arithmetic progressions

Let $\ell$ be a prime and let $1\leq a<d$ with $(a,d)=1$. A prime is $\ell$-Genocchi regular if it is not $\ell$-Genocchi irregular. Positive-density conjecture. If $\ell$ is odd, the set of $\ell$-Genocchi regular primes in the class $a\pmod d$ has positive density unless $4\ell\mid d$, the Jacobi symbol $\left(\frac{a}{\ell}\right)=1$, and $a\equiv1\pmod4$; if $\ell=2$, it has positive density unless $8\mid d$ and $a\equiv1\pmod8$. This is a weaker consequence of the preceding arithmetic-progression asymptotic conjecture; the exceptional cases are exactly those in which the relevant density constant vanishes.