There are only finitely many irregular Diophantine quadruples

A set of $m$ distinct positive integers is called a Diophantine $m$-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple $\{a,b,c,d\}$ with $a<b<c<d$ is called regular if $$d=a+b+c+2abc+2\sqrt{(ab+1)(ac+1)(bc+1)}.$$ We prove that every irr