I might be wrong (and probably the only person to know this for sure is Kostya, the sole developer of Aquamail), but I do not think that Aquamail has the notion of the "airplane mode".
From what I know, it only looks at the presence or absence of a particular type of connection (Wi-Fi, Mobile), and if configured to use that, uses, as configured.
And as far as I remember, I've used Aquamail on Wi-Fi with the mobile connectivity disabled, while traveling internationally.
So, I'd look into what else interferes.
You may want to post the information about your phone model and Android version (and if you are using any customized ROMs or non-stock battery-savers/optimizers etc.), which might provide some useful insights at what might be the culprit.