Uniqueness conjecture for the smallest element of a $D(4)$-quadruple

Let $D(4)$-$m$-tuples be sets of distinct positive integers such that the product of any two distinct elements increased by $4$ is a perfect square. Suppose that $\{a_1,b,c,d\}$ is a $D(4)$-quadruple with

$$a_1<b<c<d.$$

Smallest-element uniqueness conjecture. Then $\{a_2,b,c,d\}$ is not a $D(4)$-quadruple for any integer $a_2$ with $a_1\neq a_2<b$.

This asserts that, after the three larger elements are fixed, a $D(4)$-quadruple cannot have a different smaller positive integer in place of its smallest element. The source presents the statement as a conjecture in the context of extensions of $D(4)$-triples by smaller elements; its resolution is not specified here.