Local higher-order Fourier uniformity conjecture for the Liouville function

Let $s \geq 0$. Let $G \backslash \Gamma$ be an $s$-step nilmanifold. Let $F:G \backslash \Gamma \rightarrow \mathbb{C}$ be Lipschitz continuous and let $x_0 \in G \backslash \Gamma$. Local higher-order Fourier uniformity conjecture.

$$\int_{X}^{2X} \sup_{g \in G} \left | \sum_{x < n \leq x + H} \lambda(n) F(g^{n - \lfloor x \rfloor} x_0) \right | dx = o( H X)$$

as soon as $H \rightarrow \infty$ with $X \rightarrow \infty$. Informally, the conjecture asserts that on most short intervals, $\lambda$ does not exhibit significant correlation with any $s$-step nilsequence of bounded complexity. It is needed to establish the logarithmically averaged form of Chowla's conjecture for all numbers of shifts; the paper does not resolve it.