The difference conjecture for disjoint Golomb rulers

Let $H(I,J)$ denote the least positive integer $n$ such that there exist $I$ disjoint Golomb rulers, each a $J$-subset of $\\{1,2,\ldots,n\\}$. Difference conjecture. For integers $I$ and $J$ with $I\geq1$ and $J\geq3$, one has

$$H(I+1,J)\leq H(I,J)+J.$$

The paper states that the main conjecture would imply this inequality, but does not establish the inequality unconditionally; it also asks whether the inequality is strict when $H(I,J)>IJ$.