My Glee has had a very small amount of play as you describe from new. It was checked by my dealership and they said it was normal and demonstrated this on a brand new bike in the showroom.
I recently fitted a Cosmo forkbrace and could feel a improvement straight away but recently have felt a harshness in the feel and the front tyre, PR4 with about 6k miles, has pushed a few times on tighter bends.
I think this is a combination of the tyre wearing and me having more confidence because of the tyres and fork brace improving the feel so much over stock so I have been pushing harder.
It’s never been too bad I thought I was coming off but enough to realise I need to bring it back a notch.
Hope you get it sorted.