Kühn­el's componentwise minimality conjecture for the twisted sphere product

Let $S^3\widetilde{\times}S^1$ denote the twisted $S^3$-bundle over $S^1$, and let an $f$-vector list the numbers of faces in dimensions $0$ through $4$.

Kühn­el's componentwise minimality conjecture. The $f$-vector

$$(12,60,120,120,48)$$

is componentwise minimal among combinatorial triangulations of $S^3\widetilde{\times}S^1$.

The paper gives a $12$-vertex triangulation with this $f$-vector, while the available lower bound at the time was only $11$ vertices. Thus componentwise minimality remained conjectural.