One more try. After the WiFi sign appeared in the phone and "Launching IMAP push: account1, account2" in the notif. area shortly afterwards, I've switched off the WiFi router's power. [Since this moment, during some 10 seconds, I could see the phone's WiFi indicator's data transfer arrows flashing here and there. Apparently the OS has still not informed other applications, that the WiFi connection was really gone??] The account1 was finished with red note under its name (apparently as not synced successfully), around that time the mobile data connection (G/E/3G/H icon with flashing arrows) was established. Afterwards the account2 was synchronized successfully.
Finally a "Push mail: account1, account2" could be seen in the notif. area for a few seconds, then it went off completely.
If it matters, my Aqua is 1.4.0.53, the phone is SG Ace2, GT-I8160 with stock JellyBean 4.1.2 (by operator).