Kontsevich's sphere conjecture for dual boundary complexes of log Calabi–Yau varieties

Let $X$ be a log Calabi–Yau variety, and let $\mathbb{D}\partial X$ denote its dual boundary complex. Kontsevich's conjecture. The dual boundary complex $\mathbb{D}\partial X$ is homotopy equivalent to a sphere. This conjecture is stated as a folklore conjecture attributed to M. Kontsevich and is related to the homotopy type conjecture for character varieties; the source gives no resolution in the stated generality.