Panasonic DFD

It seems to me that only a few years ago, no camera with a narrow depth of field had good continuous autofocus for video. Even professional camcorders weren't good - they have small sensors and a wide depth of field and a lot is in focus all the time. But that makes continuous autofocus that much easier.

Then came the Canon 70D and Dual Pixel autofocus. It is pretty good with continuous focus for video. Most (all?) Canon models since then have it. Then more recently came some Sony models, I think that A6500 leads the pack, which has continuous autofocus close to the Canon.

So, I think it has only been about 3.5 years since any camera has had continuous autofocus.

Well, it looks like the GH5 is not great for continuous AF.  Perhaps better than any previous Panasonic, but not great.  And perhaps it is better with a Panasonic lens than an Olympus or other brand lens, but regardless not great.  And not great may indeed mean not usable.  Just better to manual focus.  Or stop down the lens to make the job less challenging.

BUT,  I have the 70D and don't use it for video much.  Because the quality just sucks.  I'd rather use my Panasonic LX100 and focus manually and get much better results.

BUT,  It seems to me with Sony you get a camera that overheats, and has poor menus, and you have to choose between low quality lenses, or really expensive lenses.  I'm totally not interested in Sony.

SO,  I think the GH5, or another cheaper Panasonic camera is still the best compromise for use in video unless you really value continuous autofocus over things like video quality and usability.

After all that - I don't think I really answered your question.  It may be better with Panasonic lenses, but you may just have to try out with your Olympus lenses and then decide.  Note that I don't think the Olympus has better continuous autofocus for video than Panasonic.