Are primes in arithmetic sequences equally prevalent in the limit?

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It's clear that some arithmetic sequences start off much more prime-dense than others, e.g. $6x+1$ compared to, say, $7x+13$.

As $x \rightarrow \infty$, do all arithmetic sequences even out in terms of the proportion of primes they yield, or do the ones that start out strong maintain an edge forever?

And if there is an established answer to this, does it also apply to quadratic primes, like e.g. $x^2+1$ being more prime-ful than $x^2+17$?