The density conjecture for G-irregular primes

Let ${\mathcal P}_G$ be the set of G-irregular primes, and write ${\mathcal P}_G(x)$ for the number of its elements not exceeding $x$. Let $A$ be the Artin constant,

$$A=\prod_{p\ \mathrm{prime}}\left(1-\frac{1}{p(p-1)}\right),$$

and let $\pi(x)$ denote the prime-counting function. The G-irregular-prime density conjecture. Asymptotically,

$${\mathcal P}_G(x)\sim \left(1-\frac{3A}{2\sqrt e}\right)\pi(x)\approx 0.6597765\,\pi(x).$$

This is obtained from Siegel's heuristic together with the distribution of the relevant multiplicative orders; the source gives no resolution.