I am currently experiencing a similar situation. I have 3 accounts also. 1 is Gmail, the other 2 are POP3 accounts.
The Gmail and 1 Pop3 show unread notifications as expected.
My home Pop3 account intermittently sounds a notification tone but when I check my email the message is gone from the folder.
I then check the server and the message is shown as unread. I check mail again and it does not upload the message to Aquamail. As I am sitting here right now I have about 40 unread messages on the server and none of them have been uploaded to AquaMail.
I also like this app alot but missing emails is a huge problem for me. I travel for my work and cannot miss emails.