Rational D(q)-quintuples

For a nonzero rational number q, a rationalD(q)-n-tupleis a set of ndistinct nonzero rationals {a1,a2,⋯,an}such that aiaj+qis a square for all 1⩽i<j⩽n. We investigate for which qthere exist infinitely many rational D(q)-quintuples. We show that assuming the Parity Conjecture for the twists of several explicitly given elliptic curves, the density of such qis at least 295026/296010≈99.5%.