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 \cite{btz} that every Host–Kra ${\text{F}}_p^{\omega }$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k\le p+1$. 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 $f:{\text{F}}_2^n\to \text{C}$ of large Gowers norm $\mathrm{\parallel }f{\mathrm{\parallel }}_{U^6({\text{F}}_2^n)}$ 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 $f$. A simpler version of our construction can also be used to answer a question of Candela, González-Sánchez, and Szegedy \cite{CGSS}.