Frantzikinakis--Kuca pointwise vanishing conjecture for nilsystems

Let $(G/\Gamma,\mu_{G/\Gamma},R_\alpha)$ be an ergodic nilsystem, and let $k_1,\ldots,k_\ell\in\mathbb Z$ be nonzero and distinct. For functions $f_0,\ldots,f_\ell\in L^\infty(\mu_{G/\Gamma})$, define the $U^{\ell+1}(G/\Gamma)$ seminorms as the Gowers--Host--Kra seminorms on the nilsystem. Frantzikinakis--Kuca pointwise vanishing conjecture. If $\|f_j\|_{U^{\ell+1}(G/\Gamma)}=0$ for some $j\in\{0,\ldots,\ell\}$, then

$$\lim_{n\to\infty}\int_{G/\Gamma} f_0(x)\cdot f_1(\alpha^{k_1 n}x)\cdots f_\ell(\alpha^{k_\ell n}x)\,d\mu_{G/\Gamma}(x)=0.$$

The source presents this as a consequence of the preceding Frantzikinakis--Kuca conjecture rather than as an independently named conjecture, and says that both claims are beyond the paper's scope and remain open.