Regularity propagation conjecture for disjoint Golomb rulers

For positive integers $I$ and $J$, let $H(I,J)$ denote the least $n$ admitting $I$ disjoint $J$-mark Golomb rulers in $\{1,\ldots,n\}$. A DGR is regular when $n=IJ$. Regularity propagation conjecture. If

$$H(I_0,J)=I_0J,$$

then

$$H(I,J)=IJ$$

for every integer $I>I_0$. The conjecture predicts that regularity at one value of the number of rulers propagates to all larger values; it is used in the paper as a proposed structural principle and remains unproved.