It is sufficient to clear the app from background to make the push notification not arriving.
Weird problem because not happening for every app (whatsapp, youtube is ok, smartbanking not ok or maybe arriving because whatsapp, youtube are frequently opened, have to check better...)
The solution is to go to Notifications settings and reset app preferences but after 3-4 days the issue comes back (exactly the same issue described here https://zentalk.asus.com/t5/zenfone-5-series/notifications-are-showing-only-after-opening-the-app/td... )
Asus ZenFone 4 Selfie Pro (ZD552KL) stock firmware
Some apps that automatically start in the background may occupy memory,cause system lag and drain the battery power. By managing these auto-start apps, ASUS Auto-start Manager helps free more memory, improve system performance, and save power.
Example : In Android 8 devices, go to Settings > Battery > PowerMaster > Auto-start manager. Select apps that you want to prohibit from auto-start, and then change their status from "Allow" to "Deny".
If you have configured any relevant options, please ensure that all apps receiving notifications are set to "Allow".
Thanks checked and all is fine, do not have configured any relevant options.
So far all notifications are arriving because yesterday have restored all app preferences (related to notifications), curious to count how many days needed to trigger the problem.
In addition have just received the push notification from the smart-banking app ("your ATM card have been used") ONLY when I turned on the display 😞 but can't recreate the problem at the moment since now arriving with the display turned off too, crazy problem maybe need to stay with screen off 1 hour or more
Maybe it is a "per app" problem (smart-banking app not well programmed?) but pretty sure gmail behave that way too (so I was missing emails)