The uniqueness conjecture for extensions of D(4)-triples

Let $n\neq0$ be an integer. A set of distinct positive integers is a $D(n)$-tuple if the product of any two distinct elements, increased by $n$, is a perfect square. In particular, a $D(4)$-quadruple is a set of four distinct positive integers with this property. A $D(4)$-quadruple is regular if it has the form $\{a,b,c,d_{+}\}$, where, for a $D(4)$-triple $\{a,b,c\}$ with $a<b<c$,

$$d_{\pm}(a,b,c)=a+b+c+\frac{1}{2}\left(abc\pm\sqrt{(ab+4)(ac+4)(bc+4)}\right),$$

and $d_{-}\neq0$, so that $\{a,b,c,d_{-}\}$ is also a regular $D(4)$-quadruple with $d_{-}<c$. Uniqueness conjecture for extensions of D(4)-triples. Any $D(4)$-quadruple is regular. In both the classical case $n=1$ and the case $n=4$, uniqueness of an extension of a triple to a quadruple with a larger element remains open.