Hi Jakub,
I think you’re right—this issue was very likely caused by the license, based on what happened earlier.
This morning, when I launched Visual Studio Code, before anything else appeared, IntelliPHP popped up a window with license-related information. After that, it stopped providing code predictions.
The good news is, just as you suggested, after installing the pre-release version of IntelliPHP, code prediction is working again.
Thanks so much for your help!