Unique extension conjecture for Diophantine quadruples

Let $\{a,b,c,d\}$ be a Diophantine quadruple of integers, meaning that the product of any two distinct elements increased by $1$ is a square of an integer, and suppose that $d>\max\{a,b,c\}$. Define

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

Unique extension conjecture. Then $d=d_{+}$.

Every Diophantine triple of integers is known to have the extension $d_{+}$, and the conjecture asserts that this is the unique extension larger than all three elements. It remains open; it has been proved for particular triples, including $\{1,3,8\}$, whose unique extension is $120$.