Uniqueness conjecture on simultaneous Pell equations. II

Let $A$ and $B$ be distinct positive integers. It is known that any positive solution to the simultaneous Pell equations $x^2-Ay^2=1$ and $z^2-By^2=1$ gives rise to a positive solution to the simultaneous Pell equations $x^2-(m^2-1)y^2=1$ and $z^2-(n^2-1)y^2=1$ for some distinct integers $m$ and $n$ greater than one. In this paper, we prove that the latter equations have only the positive solution $(x,y,z)=(m,1,n)$ if $\{1,b,c\}$ is a Diophantine triple with $b=m^2-1$, $c=n^2-1$ and $c\ge \max\{200b^4,2b^5\}$. Moreover, we show that the same conclusion holds if we replace the inequality assumed above by $b=\sigma p^e+1$ for some prime $p$, a positive integer $e$ and $\sigma \in\{1,2,4\}$.