In the mail client on the phone, what eats more traffic/batteries PUSH or FETCH?

Question

Email

I noticed that on my Android, TypeMail eats the most battery. I am looking for options to reduce energy consumption.
IMAP protocols. Among other things, I noticed that according to the method of receiving letters, I can choose between PUSH and FETCH. I can not understand what I need - there is not enough data.
FETCH - scheduled mail check (client-side initiative)
PUSH - notification of new messages (server-side initiative)
It's clear how FETCH works.
Connected, checked the number of UNSEEN letters, downloaded and disconnected.
It's not clear how PUSH works:
1)It seems like the client must register (tell where to send notifications, IP, etc.). But with a not very reliable connection, when changing wifi networks, when switching from WiFi to LTE and other reconnections, will it not constantly re-register? Maybe the phone needs less sleep to constantly wait for PUSH? Could it be that PUSH eats more battery than FETCH? In other words, is the battery used to maintain the PUSH channel?
2) Doesn't it happen that PUSH came, and then the client connects and downloads, as usual? That is, PUSH initializes the FETCH process.

Answer 1

Depends on several parameters.
With PUSH (actually not PUSH, but IMAP IDLE), the client maintains a persistent IMAP connection, which requires a permanent connection to the wireless network. If he is the only one who does this, then the battery will sit noticeably faster. But there is no load on the processor (until a new letter arrives).
With FETCH, the client periodically connects and wakes up the processor, there is no constant network overhead, but there are occasional bursts associated with setting up a session and checking for new messages.
It is most advantageous to use an application that is provided by the service (for example, the GMail application for Gmail) or programs like MyMail that work with any service. They do not work over IMAP, they do not require a separate permanent IMAP connection, because. they use the standard system PUSH to receive new emails.
An exception may be when you have a lot of incoming emails, for example, they arrive every minute. Then it is definitely more profitable to use FETCH.

Answer 2

Thanks to all. As a result, after switching to FETCH every half hour / hour on four accounts, the mail almost stopped eating the battery. Yes, and life has become calmer - the phone began to chime less often :-)