The incremental bound conjecture for disjoint Golomb rulers

For positive integers $I$ and $J$, let $H(I,J)$ be the least $n$ such that there are $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Incremental bound conjecture. If $I\geq1$ and $J\geq3$, then

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

The proposed inequality would give a way to construct upper bounds for $H(I,J)$ from those for smaller $I$; the paper does not prove it.