The Fourier uniformity conjecture for the Liouville function

Let $\lambda:\mathbb{N}\to\{+1,-1\}$ be the Liouville function, defined by $\lambda(n)=(-1)^{\Omega(n)}$, where $\Omega(n)$ counts prime factors with multiplicity. Let $H=H(X)$ be any function tending to infinity with $X$, and let $e(\alpha)=\exp(2\pi i\alpha)$. Fourier uniformity conjecture. One has

$$\sum_{X\leqslant x<2X}\sup_{\alpha\in\mathbb{R}/\mathbb{Z}}\left\lvert\sum_{x\leqslant n<x+H}\lambda(n)e(n\alpha)\right\rvert=o(HX)$$

as $X\to\infty$. The conjecture asserts negligible correlation between $\lambda$ and linear phases in almost all short intervals, with the frequency allowed to depend arbitrarily on the interval. It was previously known for $H\geqslant\exp((\log X)^{1/2+\varepsilon})$, and this paper proves it unconditionally for $H\geqslant\exp((\log X)^{2/5+\varepsilon})$; the conjecture remains open below the stated range.