I don't know a definite answer either way, but here's one line of questioning: I believe it is a known result that Linnik's constant is at least 2 (although it's less for almost all integers.) Do the methods used to prove that fail to distinguish between primes and composites, or between some set in which the primes have positive density and some set in which they don't?
From the other direction, however, it's apparently possible on GRH to bound the error term in the PNT in arithmetic progressions (as in Brun-Titchmarsh, although I don't think they're that tight) when N > q^{2+epsilon}. This would put some rather tricky constraints on the error term, which one could take as evidence against it.
By the way, do you have a heuristic argument that, for instance, there are infinitely many primes for which the least prime congruent to 1 mod p is at least c p^(2-epsilon)? Or even at least c p^e for some e > 1?
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