The noncommutative Hodge-to-de Rham degeneration conjecture

Let $A$ be a $\mathbb{Z}/2\mathbb{Z}$-graded, compact, smooth $A_\infty$ algebra over $\mathbb{K}$. Its negative cyclic homology $HC_*^-(A)$ is defined using the negative cyclic chain complex $\big(CC_*(A)[[u]], b+uB\big)$, where $CC_*(A)$ is the reduced Hochschild chain complex and $B$ is the Connes differential.

Hodge-to-de Rham degeneration conjecture. The negative cyclic homology $HC_*^-(A)$ is a locally free $\mathbb{K}[[u]]$-module of finite rank.

This is a fundamental conjecture of noncommutative Hodge theory, formulated by Kontsevich–Soibelman and Katzarkov–Kontsevich–Pantev. The supplied text does not state whether it has been resolved, so its status is recorded as open.