The well trim should be screwed down underneath and on the ends. There should be no way for it to come off. I think I have some pics of this if you want them.
My chrome well trim is also screwed down but a couple of screws have been pulled out (stripped). I notice the trim is also bent upwards on one side from pulling.
I'll post a couple of pics.
I bought new screws to install, but haven't had a chance to figure out how to get at them... (haven't had the top up for a long time)