This is a simple question about terminology and provenance.

I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.

I've encountered Polignac's conjecture generalizing the twin prime conjecture by replacing pairs $(p,p+2)$ with $(p,p+k)$ for any even $k$. The even more general Hardy-Littlewood conjecture actually predicts the density of any prime constellation not ruled about by residue considerations.

Does the conjecture weaker than Hardy-Littlewood merely predicting the infinite occurrence of such all such constellations enjoy a name unto itself and/or a known first appearance in the literature?

Are there other (canonical) conjectures that belong to this family, either over the integers, or other rings?