A conjecture on primes in arithmetic progressions and geometric intervals

We conjecture that any interval of the form [q^t ,q^(t+1) ], where q≥ 2 and t≥1 denote
positive integers, contains at least one prime from each coprime congruence class. We
prove this conjecture first unconditionally for all 2≤q≤45000 and all t≥1 and second
under ERH for almost all q≥2 and all t≥2. Furthermore, we outline heuristic arguments
for the validity of the conjecture beyond the proven bounds and we compare it with
related long-standing conjectures. Finally, we discuss some of its consequences.