There has been at least one post commenting that check (tick) marks have appeared against items in cultural buildings, but they don't seem to have any useful purpose. I also know that there is an old thread somewhere that complained that there used to be in indication that items were owned but not displayed in the building. This feature was lost in one of the older DLCs. It was very useful to know that you owned something but hadn't displayed it yet (maybe it was in a trading Post on another island or displayed in a different building), so at least you knew it was worth looking for it.
I suggest that these check marks could be used to restore this useful feature, for example:
Item and check mark both greyed out - You do not own the item
Item greyed out and check mark green - You own the item but it is not displayed in this building
Item highlighted I (don't care about the check mark, leave it green if it's easier) - The item is displayed in this building
EDIT. I meant to put this in General Discussion, not Product Support, because it's not a bug. Please feel free to move it if appropriate.