Existence of infinitely many empty exceptional sets

Let $E_{a,b,m}$ be the exceptional set of positive even integers in the residue class $a+b\pmod m$ that do not have a representation $n=p+q$ with primes $p\equiv a\pmod m$ and $q\equiv b\pmod m$. Empty-exceptional-set conjecture. There are infinitely many tuples $(a,b,m)$ such that

$$E_{a,b,m}=\varnothing.$$

This is presented as a plausible consequence of the preceding heuristic picture, but the authors explicitly express less confidence because precise heuristics for this assertion are more delicate. It remains open.