A Host--Kra ${\mathbf F}_2^\omega$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm

It was conjectured by Bergelson, Tao, and Ziegler that every Host--Kra ${\mathbf F}_p^\omega$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: {\mathbf F}_2^n \to {\mathbf C}$ of large Gowers norm $\|f\|_{U^6({\mathbf F}_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$.