Asymptotic chromatic-number conjecture for Kneser hypergraphs of triangulations

For an integer $r\geq 2$, let $\operatorname{KG}^{r}(\mathcal{T}_n)$ be the Kneser $r$-uniform hypergraph whose vertices are the triangulations in $\mathcal{T}_n$ and whose edges are collections of $r$ pairwise disjoint triangulations. Let $\chi$ denote its chromatic number.

Asymptotic chromatic-number conjecture. For any $r\geq 2$, there exists an integer $n_0(r)$ such that, for all $n\geq n_0(r)$,

$$\chi(\operatorname{KG}^{r}(\mathcal{T}_n))=\left\lceil\frac{n-r}{r-1}\right\rceil.$$

The conjecture is motivated by an upper bound obtained by extending the star construction, while the preceding equality does not hold for all $n\geq 3$. The supplied status evidence says the proposed equality was disproved, so the conjecture is refuted.