I normally don't use Stamina mode on my phone but recently, I had left it on by accident. Around the same time, I noticed WhatsApp Web was disconnecting itself a lot. Turns out Stamina mode does what it's meant to and disables data for WhatsApp on the phone, preventing WhatsApp Web from working properly.

It should be possible to add WhatsApp as an exception in Stamina mode to make it work, though.