I answered this question before I could finish the post, but thought it might be good to review. localStorage domains are segregated by protocol, base domain and port number. Starting in Desktop Browser with default preferences gives a random port number, so your domain could change, especially from day to day. If the port changes, you loose the localStorage, as your domain changed.
To make this work, all you need to do is the change the preference in Tools on the Advanced tab for the port number to be something other than zero. Generally port numbers above 30000 are available with a maximum of 65535.