Ramírez Alfonsín–Skałba asymptotic conjecture for primes in a numerical semigroup

Let $S_0(a,b)$ be the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi_{0,a,b}$ count the primes in $S_0(a,b)$ not exceeding $g_{0,a,b}$. Let $\pi(x)$ denote the number of primes not exceeding $x$. Ramírez Alfonsín–Skałba asymptotic conjecture. For relatively prime integers $2<a<b$,

$$\pi_{0,a,b}\sim \frac{\pi(g_{0,a,b})}{2}\quad\text{as }a\to\infty.$$

The source states that this conjecture was subsequently confirmed by Ding, Zhai, and Zhao, following an almost-everywhere result of Ding; related generalizations for $\pi_{\ell,a,b}$ are also known.