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As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.

There are a relationship between the mapping class group and the pure braid group, which the Johnson homomorphism corresponds to the Milnor's $\bar\mu$-invariant or the Artin representation into not $\operatorname{Aut}(F)$ but the automorphism group of free nilpotent quotient $\operatorname{Aut}(F/\gamma_k(F))$ where the lower central series $\gamma_k(F)$ of a free group.

(It is known that the image of the pure braid group under the injective Artin presentation into $\operatorname{Aut}(F)$.)

Then, is it also unknown that what is the image of the pure braid group under the Artin presentations or the Milnor invariants?

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    $\begingroup$ I don't manage to parse the grammar of this question. Perhaps someone who upvoted it can help out? $\endgroup$
    – Stefan Witzel
    Commented Jun 24, 2021 at 20:54

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