Chen's Romanov-type lower-density conjecture

Let $\mathscr{A}$ and $\mathscr{B}$ be two sets of positive integers. For a set $S$ of positive integers, write $S(x)=|S\cap[1,x]|$. Chen's Romanov-type conjecture. If there exists a constant $c>0$ such that

$$\mathscr{A}(\log x/\log 2)\mathscr{B}(x)>cx$$

for all sufficiently large $x$, then the set

$$\{2^a+b:a\in \mathscr{A},~b\in \mathscr{B}\}$$

has positive lower asymptotic density. The conjecture was disproved by the paper, which constructs a concrete counterexample; it was one of two Romanov-type conjectures posed by Chen in 2008.