The regularity conjecture for Diophantine quadruples

Let $\{a,b,c,d\}$ be a Diophantine quadruple, meaning that each of $ab+1$, $ac+1$, $ad+1$, $bc+1$, $bd+1$, and $cd+1$ is a perfect square. Suppose that $ab+1=r^2$, $ac+1=s^2$, and $bc+1=t^2$, and define

$$d_+=a+b+c+2abc+2rst.$$

The regularity conjecture for Diophantine quadruples. If $d>\max\{a,b,c\}$, then $d=d_+$. All known Diophantine quadruples were regular when this conjecture was formulated, and the source describes it as a stronger version of the nonexistence theorem for Diophantine quintuples that remained open there.