Trouble I have created a filters for my custom post type universities using ajax

I have created a filters for my custom post type universities using ajax.
I have used a shortcode and then getting the result using the ajax request. The filters are working fine on universities page.
But I want to search for universities from homepage using a form and passing arguments to the universities page. When clicking on submit the page redirects and it shows a 404 not found error. While when refreshed is it redirected to universities page but without the list of arguments.
I think there may be a way to pass arguments to the universities page. I used get_posts() on universities page to query result.
May be query_var. I am not sure. A little help or guidance will be very helpful.
Regards

The issue names no repository files, tests, or reproducible plugin behavior and instead describes a site-specific WordPress form flow. Start with the linked homepage and universities-page flow, then clarify the expected URL and query parameters; done would require a defined plugin-level failure and a specified change.