The default setting isn’ t ideal for this. You should set
"julia.persistentSession.enabled": true,
"julia.persistentSession.tmuxSessionName": "julia_vscode_${workspaceFolderBasename}",
to use project REPLs.
If you use a server and set this. The need to restore should be minimal.
And my branch of the vscode extension Looking for opinion on dot methods completion implementation contains saving and restore of inline results on switching devices that connect to the remote server.