The persistence conjecture for regular disjoint Golomb rulers

Let $H(I,J)$ be the least positive integer $n$ such that there exist $I$ disjoint Golomb rulers, each a $J$-subset of $\\{1,2,\ldots,n\\}$. A DGR is regular when $H(I,J)=IJ$. Persistence conjecture. For $I_0\geq2$, if

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

then

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

for every integer $I>I_0$. The paper presents this as a consequence of the main conjecture, and reports some verified cases, but leaves the general assertion unproved.