Categorical Torelli theorem for hypersurfaces

Let $X\subset {\mathbb{P}}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component ${\mathcal{A}}_X$. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines $X$ uniquely if $d>3$ or if $d=3$ and $n>3$. This generalizes a result by D. Huybrechts and J. Rennemo, who proved the same statement under the additional assumption that $d$ divides $n+2$.