The stronger Diophantine quadruple conjecture

A Diophantine $m$-tuple is a set of $m$ distinct positive integers $\{a_1,\dots,a_m\}$ such that $a_i a_j+1$ is a perfect square for every pair of distinct indices $i,j$. Let $\{a,b,c,d\}$ be a Diophantine quadruple and define

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

Stronger Diophantine quadruple conjecture. If $\{a,b,c,d\}$ is a Diophantine quadruple and $d>\max\{a,b,c\}$, then $d=d_+$.

The source presents this as a stronger version of the conjecture that no Diophantine quintuple exists. Its resolution status is not specified and remains open.