Lutz–Kühnel's minimality conjecture for tight triangulations

Let $M$ be a manifold, and let $C$ be a tight triangulation of $M$. The $f$-vector of $C$ records the number of faces in each dimension.

Lutz–Kühnel minimality conjecture. Tight triangulations of $M$ minimize every entry of the $f$-vector among all triangulations of $M$.

The claim is trivially true in dimension two and was proved in dimension three by Bagchi, Datta and Spreer. The source gives no resolution in general dimensions.