The generalised Goldbach conjecture for weighted prime sums

Let $m_1$ and $m_2$ be positive integers. Write $d=\gcd(m_1,m_2)$, and let $2^s$ be the largest power of $2$ dividing both $m_1$ and $m_2$. For a sufficiently large integer $n$, impose

$$\gcd(n,m_1)=\gcd(n,m_2)=d,$$

and

$$n\equiv m_1+m_2\pmod{2^{s+1}}.$$

Generalised Goldbach conjecture. Under these conditions, there exist primes $p$ and $q$ such that

$$n=m_1p+m_2q.$$

This conjecture includes the even Goldbach conjecture when $(m_1,m_2)=(1,1)$ and Lemoine's conjecture when $(m_1,m_2)=(1,2)$. Its general proof is stated to be out of reach; the paper empirically tests instances for coefficients up to $40$.