A counterexample to the Gowers $U^k$ uncertainty conjecture for $k\ge 6$

The Donoho–Stark uncertainty principle asserts $N^d\le \mathrm{\mid }E\mathrm{\mid }\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }$, and the additive energy uncertainty principle of Aldahleh–Iosevich–Iosevich–Jaimangal–Mayeli–Pack refines it to $1\le \mathrm{\mid }E\mathrm{\mid }\cdot \mathrm{\parallel }{1}_{\mathrm{\Sigma }}{\mathrm{\parallel }}_{U^2}^{4\mathrm{/}3}$. It has been conjectured that the natural higher-order analogue $$1\le \mathrm{\mid }E\mathrm{\mid }\cdot \mathrm{\parallel }{1}_{\mathrm{\Sigma }}{\mathrm{\parallel }}_{U^k}^{2^k\mathrm{/}(k+1)}$$ holds for every $k\ge 2$, the coset case giving equality. We show that this fails for every $k\ge 6$. The counterexample is the two-point signal $f={\delta }_0-{\delta }_1$ on ${\mathbb{Z}}_3$, for which the left-hand quantity equals $2\mathrm{/}3(2(k+1){)}^{1\mathrm{/}(k+1)}$; this is $<1$ precisely when $k\ge 6$, and tends to $2\mathrm{/}3$.