Kühnel and Lutz's tightness conjecture on strong minimality

Let $X$ be a simplicial complex. Call $X$ $\mathbb{F}$-tight when it has the tightness property over a field $\mathbb{F}$, and call it strongly minimal when it achieves the componentwise smallest face vector among all triangulations of its geometric carrier $|X|$. Kühnel and Lutz's tightness conjecture. Every $\mathbb{F}$-tight simplicial complex is strongly minimal. If true, this would strengthen ordinary minimality by asserting that tight triangulations minimize every face-number component among triangulations of the same carrier; the source presents it as an open conjecture.