PUSH & Receiving mails & Sync & 15000+ emails = Troubles

Hello! I hope this research will be able to help people in such a troubles as mine and Super hope for Developer comments!

I have simply standard conditions:

  1. Mail box on Yandex.ru (Russian IT monster, it is not a no name company with working wrong products)
  2. 15000+ emails
  3. I need have a good search in all emails, that means full offline (downloaded) data (mails & attaches)
  4. I need to receive emails as soon as they arrive in my box, as I understand it is PUSH or 1-minute sync in settings if PUSH works not well.
    Simple, yes? I just want to use emClient!

What I have:

  1. PUSH does not work, it works only when I have any activity (sending, receiving, moving, marking) with any other mails in current box. As soon as I keep EM in background (not minimize, not close) and work in another app, PUSH does not work at all. I come morning an there no changes even 10 emails are in my box, they did not received by PUSH.
  2. If I choose Menu - Settings - General - Sync every 1 min, that option works with troubles because syncing takes a lot time. Near 1 min (31 sec if no new emails). And it rejects after 1 min and starts from beginning. Sometimes need to restart EM to get emails. Because they exist on server but do not receiving by EM.
  3. BTW! Another mail clients (will not share names to be polite) works with PUSH ok.

As result I receive not all emails in not correct time.

Trust me, I have tried all. No antivirus, no firewall, static IP for sure and much more.

Finally solution has been found!

As soon as I sync all data, download all my emails with content and attaches, I turned OFF checkbox in SYNC field in IMAP settings every account about downloading messages for use them offline. Every new incoming email I open any way and it any way downloads to me after opening.

What result I got after uncheck this option in IMAP setting? Syncing now takes less than 1 sec!!! Before 31 sec!

May be any Developer can give any comment about it? I use PRO license btw.