I just opened a project which had the Browser open.
I closed the browser, saved the project and re-opened the project and the Browser remained closed.
This was with the screenset unlocked
I then experimented with locking the screenset with the Browser open, hid the browser, saved, closed & re-opened and the Browser reverted to being open.
This is exactly how it should be so at this moment, I don't know what's causing your problem
With an unlocked screenset, saving the project should honour the visibility of not only the Browser but everything else you can see when you save