Kontsevich–Soibelman noncommutative Hodge structure conjecture

Let $X$ be a smooth and compact noncommutative space over $\mathbb{C}$. The periodic cyclic homology $HP_{\bullet}(C_X)$ is defined from the category $C_X$ associated with $X$, and a $\mathbb{Z}/2$-grading is said to refine to a $\mathbb{Z}$-grading when the latter grading comes from an integral grading. Noncommutative Hodge structure conjecture. The periodic cyclic homology $HP_{\bullet}(C_X)$ carries a natural functorial pure $\mathbb{Q}$-noncommutative Hodge structure with regular singularities. Furthermore, if the $\mathbb{Z}/2$-grading on $X$ can be refined to a $\mathbb{Z}$-grading, then the noncommutative Hodge structure on $HP_{\bullet}(C_X)$ is an ordinary pure Hodge structure, i.e. belongs to the essential image of the functor $\mathfrak{N}$. This is presented as the main conjecture for smooth and compact noncommutative spaces; the supplied text gives no resolution or partial result establishing it.