Exceptional-set growth conjecture for Goldbach representations in progressions

Assume $m$ is even, and let $(\mathbb Z/m\mathbb Z)^\times$ denote the invertible residue classes. For admissible $a,b\in(\mathbb Z/m\mathbb Z)^\times$, let $E_{a,b,m}$ be the set of positive even $n\equiv a+b\pmod m$ that cannot be written as $n=p+q$ with primes $p\equiv a\pmod m$ and $q\equiv b\pmod m$. Define

$$E_{\max}(m):=\max\{n\in E_{a,b,m}:a,b\in(\mathbb Z/m\mathbb Z)^\times\}$$

and

$$L_{\mathrm{avg}}(m):=\frac{1}{\phi(m)^2}\sum_{a,b\in(\mathbb Z/m\mathbb Z)^\times}|E_{a,b,m}|.$$

Exceptional-set growth conjecture. As $m\to\infty$,

$$E_{\max}(m)=O\bigl(m^2(\log m)^2\bigr),\qquad L_{\mathrm{avg}}(m)=O(m^\varepsilon)\quad\text{for every }\varepsilon>0.$$

The conjecture is motivated by heuristic arguments and numerical data, which suggest roughly quadratic growth for $E_{\max}(m)$. Analytic methods in the paper do not establish even finiteness of all the exceptional sets, so these bounds remain open.