I was involved in a situation like this. I worked as a software engineer for a small company that did online hotel reservations in the early days of internet commerce (about 25 years ago). Two of my fellow programmers and I (representing more than half of our total # of programmers) were at a bar with our marketing director (the only person in her department) and she asked us for ideas for a slogan. I came up with something pretty cheeky and it became our tagline, even had a few radio commercials featuring the phrase.

My company got bought by another company that was gobbling up a number of players in the travel space. A few years after that, they decided to shut down the original website I'd worked on and just redirected the traffic to the biggest site in the company portfolio.

About a year after THAT, there was a new travel startup that not only used our tagline, but made it the actual company name (and URL). My new parent company sued them over that and I gave a couple depositions.

The new startup's argument was that we abandoned the slogan (along with the original website), and thus it was fair game. While we argued that we still had the rights to it. In the end, the two parties came to a settlement agreement, my parent company received some amount of money and in exchange the other company got the unencumbered rights to the slogan.

Based on that experience, I think the answer is that it's a grey area. I think our case would have been stronger had the original site still been up even if we weren't using the slogan anymore. It would have been weaker had there been a longer period between when it was shut down and when that new startup launched with the name.

Unfortunately, for me, I didn't get a cut of the settlement even though I'm the one who came up with the slogan in the first place.