I have recently had this issue on our domain.
We disabled offline files on folder redirection last week, and since then I have had a few users with 'green ticks' on shortcuts.
I originally put it down to the fact that it meant they were connected to their files on the NAS drive with sufficient bandwidth as in some cases i.e laptop users were not conected to the VPN the icons would get the grey crosses. Yesterday however a user
raised a ticket where their overlays would change at random intervals from green ticks, to sharing, to admin only and crosses.
It turns out that despite disabling offline files and changing the GPO to not automatically offline redirected folders, the cache was still in effect. So we cleared the cache by adding the DWORD FormatDatabase to HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\services\CSC\Parameters
with a value of 1 and then rebooting. This has solved our problem.
Hope this helps someone out.
Please note if you are using offline files in the same way that we were, please make sure that you have synced your files before adding the above DWORD as any unsynced files will be lost.