Hi,
Thank you, there is indeed an issue there. I tested on the back-end preview window, where it works perfectly. Then I investigated the page source, and found that there is an additional element injected as a parent to the date picker field, which should not be there: https://i.imgur.com/dzGl43p.png
The source of that ijection is a custom script within the site footer: https://i.imgur.com/ziNPnxs.png
I strongly recommend removing that, as it changes the behavior of that field, and the search script cannot properly update the field values.