What is the menu parameters for /janus? What page is it trying to navigate to. Same for /pages/14-janus. What is it trying to navigate to? I need the raw URLs without SEF to understand where the menu is trying to go. Sounds like just an access denied redirect in GJ maybe pointing to the wrong page causing a loop. Does it still loop with SEF off?

I believe I see the issue. The redirect has the same Itemid as the category menu item. The category menu item parameters attempt to populate again during routing processing. The routing thinks you're then trying to access a category. This is what is causing the infinite loop.

I don't see why it should be doing that though. Does the user not have access to that category? There's no reason GJ should attempt to redirect away from a category if they've access to it. So for example if the category is set to Registered then your menu item should also be set to Registered. We need to first address why it's even redirecting to begin with.

Unfortunately I do not have time to investigate this further right now, but have added a bug ticket to take a look for a future release.

forge.joomlapolis.com/issues/7911

Is your install completely up to date with latest builds? Also ensure CBs system plugin in Extensions > Plugins is published, public, and ideally ordered first. The only other thing I can think of is for some reason the menu parameters are not pushing to GET, which our system plugin handles.

Even if I set all categories to public, then the links are wrongly built by GJ

Your links are all correct and valid. The problem is likely that "func" isn't being treated right in the presence of "id", which is being given to GJ by the menu item GET variables.

Unfortunately like I said above I don't have time to investigate this right now. The routing has to account for too many usecases and is likely what is causing these problems so something might have to be deprecated soon to stop confusing the router. The easiest solution is to just don't create menu items for specific categories or groups for the time being or if you do so use Joomla external menu items to link to them.