This is VERY disconcerting.
Permissions inside the PlexMediaShared folder will only break when:
- Media is placed there (manually or by automated tools)
- Plug-ins manually installed incorrectly.
I can give you the commands for Task Scheduler to fix it but would like to know how it got broken because DSM itself maintains the top-most level control.
Given the date/time of the logs is way off, I believe preventative maintenance should be done to return the shared folder to a known-good state.
Here’s the FAQ to confirm PlexMediaServer permissions (inside the shared folder) are correct.