The uniqueness conjecture for extending Diophantine triples

Let $\{a,b,c,d\}$ be a Diophantine quadruple of integers, and suppose that

$$d>\max\{a,b,c\}.$$

Define

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

Uniqueness conjecture for extensions. Then

$$d=d_{+}=a+b+c+2\left(abc+\sqrt{(ab+1)(ac+1)(bc+1)}\right).$$

This is presented as the stronger version of the folklore conjecture that every Diophantine triple can be extended to a quadruple with a larger element in a unique way. The source states that this stronger conjecture remains open.