The common-largest-elements conjecture for D(4)-triples

Let $\{a_1,b,c\}$ and $\{a_2,b,c\}$ be $D(4)$-triples with

$$a_1<a_2<b<c.$$

Common-largest-elements conjecture. Then $\{a_1,a_2,b,c\}$ is a $D(4)$-quadruple.

This conjecture asserts a uniqueness-type property for extending $D(4)$-triples with the same two largest elements. The paper states that it is supported by two theorems, but the general assertion remains open.