A note on the Cramér-Granville model

We show the existence of a set $A\subseteq {\mathbb{Z}}_{\ge 2}$ satisfying the estimates of the Bateman–Horn conjecture, Goldbach’s conjecture, and also $$\mathrm{\#}\{p\le x\text{ prime }\mathrm{\mid }p\in A\}\gg x(\log \log x)\mathrm{/}(\log x{)}^2\mathrm{.}$$