Posting the results:
@jocaferlo has a desktop mail app checking same accounts at his office computer, and wanted to avoid double notifications.
He was using Tasker to turn push mail on and off, driven by the device connecting to a WiFi network at the office.
In the log, I could see that Tasker was telling AquaMail turn push on and and off close to 100 times a day -- not the handful of times as would be expected.
I don't know if this is a bug in Tasker, or a compatibility issue between the Android device and the office WiFi network (more likely), but the result was --
-- AquaMail was getting told to start up push mail about 50 times a day.
My recommendation was to use Tasker to only turn off notifications when at the office.