The conjecture on one-element extensions and deletions of difference sets

Let $G$ be a finite group, and let an ADS be obtained from a difference set in $G$ by adding or removing one element. The one-element ADS conjecture. If $G$ is cyclic or has odd order $v$, then no other difference sets form an ADS by adding or removing an element. The conjecture is motivated by the exhaustive examples described in the paper, which occur in $2$-groups; the source gives no resolution status.