Kühnel–Lutz conjecture on tight triangulations and strong minimality

Let $\mathbb{F}$ be a field. A simplicial complex $X$ is $\mathbb{F}$-tight if it is connected and, for every induced subcomplex $Y$ of $X$, the inclusion-induced map

$$H_{\ast}(Y;\mathbb{F})\to H_{\ast}(X;\mathbb{F})$$

is injective. A triangulated closed manifold is strongly minimal if, for every triangulation $Y$ of its geometric carrier, its face numbers are no larger than those of $Y$ in every dimension. Kühnel–Lutz conjecture. For any field $\mathbb{F}$, every $\mathbb{F}$-tight triangulated closed manifold is strongly minimal. The conjecture is known in dimensions at most $3$ by the results of this paper; its validity in higher dimensions remains open.