Thank you!
Turns out this was a known problem. We recently moved the asset folder for ajax search pro, and for some reason on sg-host staging environments the URL of the wp-content directory is reported with the non secure protocol “http://” instead of “https://”.
It’s not a big deal, usually WordPress auto-corrects that at some point, but here it failed. I had to add a single line of code to force the correct protocol, now it should be fine.