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I'd like to see a similar effort, but to replace Grammarly (i.e. make it private, secure, and offline)

Actually, the winner of the shared task on grammatical error correction https://www.cl.cam.ac.uk/research/nl/bea2019st/ is https://neural.mt/papers/edinburgh/bea19.pdf which uses the same machine translation codebase, Marian, as the Bergamot does.