On a conjecture of Ramírez Alfonsín and Skałba II

Let $1<c<d$ be two relatively prime integers and $g_{c,d}=cd-c-d$. We confirm, by employing the Hardy–Littlewood method, a 2020 conjecture of Ramírez Alfonsín and Skałba which states that #\left\{p\le g_{c,d}:p\in \mathcal{P}, ~p=cx+dy,~x,y\in \mathbb{Z}_{\geqslant0}\right\}\sim \frac{1}{2}π\left(g_{c,d}\right) \quad (\text{as}~c\rightarrow\infty), where $\mathcal{P}$ is the set of primes, ${\mathbb{Z}}_{⩾0}$ is the set of nonnegative integers and $\pi (t)$ denotes the number of primes not exceeding $t$.