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. (arXiv:2303.04853v1 [math.DS]) http://arxiv.org/abs/2303.04853