@S1m The other way round (making XMPP apps wake up upon receiving WebPush notifications) is currently not standardized. We are talking about a Push Notifications 2.0 XEP and I have been advocating for using WebPush but I have also met some resistance because XMPP server devs aren't super keen on a hard dependency to HTTP.
And personally I find the other direction, making the XMPP client a distributor, much more interesting.