M\"obius Disjointness for Nilsequences Along Short Intervals

For a nilmanifold $G\mathrm{/}\mathrm{\Gamma }$, a $1$-Lipschitz continuous function $F$ and the M"obius sequence $\mu (n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac{1}{HN}\sum _{n\le N}\mid \sum _{h\le H}\mu (n+h)F(g^{n+h}x)\mid$$ as $H,N\to \mathrm{\infty }$. The bound is uniform in $g\in G$, $x\in G\mathrm{/}\mathrm{\Gamma }$ and $F$.