Rational Diophantine sextuples with strong pair

A set of $m$ distinct nonzero rationals $\{a_1,a_2,\dots ,a_m\}$ such that $a_i{a}_j+1$ is a perfect square for all $1\le i<j\le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le m$, then we say the $m$-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.