I’m sorry that you’re having trouble. I’ll warn you in advance that I don’t have confidence that this is going to be a helpful post
I have a pair of LSX IIs as well as two pairs of LS 50 IIs. All three pairs have worked flawlessly once on appropriate firmware versions.
You’re on 1.3, which is the latest. So no firmware issue.
I have seen reports of people having issues while on wireless. You don’t mention how your speakers are connected. Are they wireless?
There have also been reports of people having issues with specific vendor’s ethernet switches - I know for certain people have had issues with TP-Link switches. Possibly other vendors as well. Since the LSX IIs are so small, I recommend you try a wired connection on the same switch or router that your core is connected to. You can do this with just the primary speaker to test it out. That may not be a long-term solution but at least it’s a test you can do to help figure this out.
Sorry I can’t point my finger directly at the issue and help get you working but maybe there’s something here that’ll help.
[Edit : Here’s a thread that includes people discussing TP-Link as a potential culprit behind an issue similar to yours though it pre-dates KEF’s Roon Ready implementation so may actually not be relevant: KEF Roon Tested disappears and leaves only Airplay option]
Is the WiFi option you’re referring to the “Wi-Fi” option in the picture below? If so, the name of that button is always Wi-Fi. It doesn’t change based on whether you are cabled or on wireless. It always acts as the network input.