The plugin does support SSL of course, it's most likely only the caching causing the issue.
Try to open the search settings and save them again, it will regenerate and re-cache the plugin CSS content.
I checked your site, but I see only one warning on the console regarding an icon. If you save the options, the icon path will be re-generated with the https: protocol.
The other issues are not present on my side, it's most likely your browser or website cache. If you save the search options and refresh your site with CTRL + SHIFT + R then the issues should disappear.Best,
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