Ternary Goldbach conjecture with primes in one progression modulo $3$

Let $p,q,r$ be primes. The ternary progression Goldbach conjecture. Every odd integer $n>5$ can be written as

$$n=p+q+r,$$

with

$$p\equiv q\equiv2\pmod3.$$

This is stated as a ternary analogue of one of the explicit mod-$4$ refinements. The paper notes that results for the binary case would also yield ternary consequences, but this particular progression statement remains unproved.