No Diophantine sextuples in imaginary quadratic integer rings

Let $K$ be an imaginary quadratic number field and let $\mathcal O_K$ be its full ring of integers. Let $A\subseteq\mathcal O_K\setminus\{0\}$ be finite. Suppose that, for every pair of distinct elements $a,b\in A$, there exists $r\in\mathcal O_K$ such that $r^2=ab+1$. Then

Equivalently, no full ring of integers of an imaginary quadratic field contains a Diophantine sextuple. The bound is uniform over all imaginary quadratic fields, including $K=\mathbb Q(\sqrt{-d})$ with $d>0$ squarefree and $d\equiv3\pmod4$, for which $\mathcal O_K$ strictly contains $\mathbb Z[\sqrt{-d}]$. The case $ab+1=0$ is allowed, with $r=0$.