Can you please include screenshots of the updates you are referring to?
Collection projects like the one in your screenshot do not create notifications (upper right red “conversation bubble”) and the only dashboard/homepage updates are journal posts and changes to the project. There are only two journal posts on that project. Changes to the project would include whether the project admin/managers changed or if the project admin turned on “Trust”, which doesn’t seem to be the case here.
Examples of the type of screenshots that would be helpful, if they indicate updates from this project:
Upper right notifications:
I would also be interested to see if you are “subscribing” to anything that might account for the issue you are experiencing https://www.inaturalist.org/subscriptions
e.g. under Projects https://www.inaturalist.org/subscriptions?type=project
Places https://www.inaturalist.org/subscriptions?type=place, or
(You can embed screenshots directly here on the forum, no need to upload elsewhere)