Extension conjecture for regular disjoint Golomb rulers

Let $I$ and $J$ be positive integers. A regular $(I+1,J,(I+1)J)$-DGR is a collection of $I+1$ disjoint $J$-mark Golomb rulers partitioning $\{1,\ldots,(I+1)J\}$. Regular extension conjecture. For every Golomb ruler $A_1\subseteq\{1,2,\ldots,(I+1)J\}$ with $|A_1|=J$, if $H(I,J)=IJ$, then there exists a regular $(I+1,J,(I+1)J)$-DGR containing $A_1$. This strengthens the preceding regularity idea by requiring extension of every prescribed ruler; it is stated as a conjecture without a proof.