I've never used MS' e-mail, so I don't know. I know that MS introduced IMAP for their e-mail services very late, in the Fall of 2013. But AFAIR, they didn't have "IDLE" enabled at that point. (And I am not sure if they every introduced it. -- I wouldn't be surprised if they've never enabled it.)
These references support this fact:
https://groups.google.com/forum/#!topic/maildroid/1Ec-wlJeoughttps://blogs.office.com/2013/09/12/outlook-com-now-has-imap/I am sure Kostya knows it better.
Also, - you may want to make sure that you have "Push" enabled in the following places in the configurations:
1) Settings -> Mail Receiving -> IMAP push
2) [Account] -> Options and Folders -> Account options -> IMAP Push
3) [Account] -> Options and Folders -> IMAP options -> IMAP Push session duration (set to the desired value)
4) [Account] -> Options and Folders -> Folders -> [Folder Name] => (if configured to be synced, e.g. Inbox, Drafts, ... ): "Use IMAP Push".
In your case, [Folder Name] is Inbox.
[Account] - means the long-touch on the account name in the account list.
HTH.