Improved bounds for the Fourier uniformity conjecture

Let $\lambda$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{\alpha \in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} \lambda(n) e(n\alpha)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H = H(X) \geq \exp((\log X)^{2/5+\varepsilon})$. This improves upon a result of Walsh towards the Fourier uniformity conjecture.