Have you tried using cookies or local storage? Those may be places you can stash user data that you hope to be persistent. Local Storage vs Cookies
Alternatively, if you have a user that allows cookies, you could always use server side resources-- like a session or a DB entry keyed off the CSRF*. Session Variables vs Database Storage in Terms of Speed
*My multi-page forms are always on one actual page in one set of form tags. Each 'page' is just a section that is shown via CSS-- that still works if javascript is off, it is just one looong form. So from 'page' to 'page' of the form my CSRF stays the same. Also, since the user never navigates away, 'page' navigation doesn't 'kill' any data entered into the form.
What I ended up doing was running some JavaScript that made a REST request after the DOM loaded and manipulated the form elements according to the elements and their properties/attributes, which made the page elements match what I had stored. Previous to that, I had used PHP to echo PHP variables server side into my JavaScript that ran client side-- but that task really is better suited to front end code. It wasn't pretty, but it worked for presetting dropdowns.
I stumbled on someone else's representative example of how I first manipulated a form element with PHP/JavaScript: Injecting Stored Information into JavaScript via PHP In the example, you can see how to put information you have stored into your JavaScript. This felt messy and unprofessional to me, so I do things differently now. Now, I do it all in PHP by setting the proper 'selected' attribute in the option tag-- obviously you'll need to edit it for other elements via the text value, checkbox checked, or node text for textarea elements. For AJAX requests, the selected option is indicated in the return data and handled in a standardized way-- even if it is just my own standard.
Why am I telling you this server side stuff that may be irrelevant-- because you may want to consider storing the in progress form data server side which you have much control over...