The main conjecture on arbitrary disjoint Golomb rulers

For positive integers $I$ and $J$, an $(I,J,n)$-DGR is a collection of $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Let $H(I,J)$ be the least positive integer $n$ for which an $(I,J,n)$-DGR exists. The main DGR conjecture. If $A$ is any set of positive integers with $|A|=H(I,J)$, then $A$ contains $I$ disjoint Golomb rulers, each a $J$-subset of $A$. This generalizes the conjecture of Komlós, Sulyok and Szemerédi in the case $I=1$ and is the principal conjecture of the paper; no resolution is supplied.