The arithmetic progression nonvanishing conjecture for Fermat quotients

For a prime $p>2$, write $q_2(p)=(2^{p-1}-1)/2$. Arithmetic progression nonvanishing conjecture. For every pair of positive integers $m>a>0$ with $\gcd(m,a)=1$, there are infinitely many primes $p\equiv a\pmod m$ such that

$$q_2(p)\not\equiv0\pmod p.$$

This asserts nonvanishing of the Fermat quotient of $2$ in every reduced arithmetic progression. The source says it is believed to be true but gives no proof or resolution.