Doesn't Work on Safari Private Mode

When I open a notebook or `jupyter lab` in Safari private mode, it invariably tells me that the workspace is already in use, and I need to pick a new name, etc. No name is accepted.

When I switch to the non-private mode, this problem clears up.

I suspect this has to do with cookie storage or something from private mode.

Reproduce the issue by opening a notebook or running `jupyter lab` in Safari private mode, then compare the workspace behavior with normal mode. Investigate the workspace-in-use and name-selection flow, and consider the reported cookie-storage difference; done means a private-mode session opens without rejecting every workspace name.

Written by the indexing model from the issue text.