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No, that won't work. The webmail uses a simple custom protocol.

Perhaps someone can rewrite it using JMAP in the future...

JMAP seems opinionated and somewhat limiting as an API, migrating to it from a more featureful/ergonomic API might be a pain.

What do you mean by JMAP is 'limiting as an API'?

It actually lifts the IMAP protocol to be used by web apps easily.