Tao's nilmanifold formulation of the logarithmic Sarnak conjecture

Let $s\in\mathbb{N}$, let $G/\Gamma$ be an $s$-step nilmanifold, where $G$ is a connected, simply connected $s$-step nilpotent Lie group and $\Gamma$ is a cocompact discrete subgroup, let $F:G/\Gamma\to\mathbb{C}$ be Lipschitz-continuous, and let $x_0\in G/\Gamma$. For $g\in G$ and $h\in\mathbb{N}$, write $g^h x_0$ for the induced action on the nilmanifold. Tao's nilmanifold conjecture. One has

$$\lim_{H\to+\infty}\limsup_{N\to+\infty}\mathbb{E}^{\operatorname{log}}_{n\leq N}\sup_{g\in G}\left|\mathbb{E}_{h\leq H}\mu(n+h)F(g^h x_0)\right|=0.$$

Tao's result identifies this nilmanifold statement as equivalent to the logarithmic Sarnak conjecture. The paper uses this formulation to connect logarithmic Möbius disjointness with polynomial complexity; the general assertion remains open.