Equidistribution of prime values over allowable congruence classes

Let $m\ge2$ be an integer, and let $f(X)\in\mathbb Z[X]$ represent infinitely many primes. A congruence class modulo $m$ is allowable if every integer $r$ in it satisfies $\gcd(f(r),m)=1$. Equidistribution conjecture. The integers $n$ for which $f(n)$ is prime are asymptotically equidistributed over the allowable congruence classes modulo $m$. This weaker conjecture is implied by Hardy–Littlewood's Conjecture F and is used to control the residue classes of prime values; it remains open in the stated generality.