Barilliance detects when the site is using the HTTPS protocol and will automatically serve all Barilliance generated content over HTTPS. This includes but not limited to:
The Barilliance script itself
Recommendation images
Recommendation links
Images used as close button in popups
images served in Show Banner rules
etc...
In most cases when switching over to HTTPS things will "just work" however you should still notify Barilliance before switching over so we can review if any special customizations made for your website rely on content that is only available via HTTP
Additionally you should review the following points:
Verify if image URLs in your product feed will work when changing the protocol to HTTPS
Update URLs added inside popups, message bars, embedded forms etc... to HTTPS and test that they are working correctly
Update cart URLs in email template to HTTPS (in most cases the HTTP version will redirect to HTTPS so links will still work but often during the redirect tracking parameters will be stripped making it impossible to track performance and clicks)