Yes, or at least, I think so.
If my understanding is correct, since the web browser is "pretending" that the HTML file you're loading is from a website... If you run the HTML file from the same folder (this includes the file path) and use the same filename (between different versions, i recommend just changing the name to AccidentalMind.html or similar), the browser should detect it as the same "website" allowing you to access the same local storage.
However, if something should happen and the local storage gets cleared out, you will loose all progress and have to start again, so making external saves is something that you should probably do anyway just to make sure the browser doesn't eat your saves by accident.