Levit–Mandrescu unimodality conjecture for pure flag complexes

Let $Delta$ be a $(d-1)$-dimensional pure flag complex on $2d$ vertices, and let its $f$-vector be the vector counting faces by dimension. Levit–Mandrescu conjecture. The $f$-vector of $Delta$ is unimodal. This conjecture concerns the face enumeration of pure flag complexes and is stated here in the context of independence complexes of very well-covered graphs; the source does not indicate whether it has been resolved.