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I think not. Consider these constellations $12,18,12,18$ and $18,12,18,12.$ The first can never have the first prime $3 \bmod 5$ but $1,2,4 \bmod 5$ are not ruled out. Swap the roles of $2$ and $3$ for the second. So those I would expect to have roughly equal frequency up to $x.$ On the other hand $12,12,18,18$ rules out the first prime being $1,3 \bmod 5$ but $2,4 \bmod 5$ are not ruled out 0`1 . So I would expect it to occur only $\frac23$ as often as the first two.