  For a nonzero rational number $q$, a rational $D(q)$-$n$-tuple is a set of
$n$ distinct nonzero rationals $\{a_1, a_2, \dots, a_n\}$ such that $a_ia_j+q$
is a square for all $1 \leqslant i < j \leqslant n$. We investigate for which
$q$ there 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 $q$ is at least $295026/296010\approx
99.5\%$.
