Kalai–Meagher conjecture on intersecting families of triangulations

Let $d4f4_n$ denote the set of triangulations of a convex $n$-gon, and let $d4f4_{n-1}$ denote the corresponding set for a convex $(n-1)$-gon. The independence number $d6fc(\operatorname{KG}(d4f4_n))$ is the size of the largest family of triangulations in $d4f4_n$ such that any two triangulations intersect.

Kalai–Meagher conjecture. For every $n\geq 3$,

$$\alpha(\operatorname{KG}(\mathcal{T}_n))=|\mathcal{T}_{n-1}|=C_{n-3},$$

where

$$C_k=\frac{1}{k+1}\binom{2k}{k}$$

denotes the $k$th Catalan number.

The conjecture asserts that the largest intersecting family consists of all triangulations containing a fixed diagonal, such as ${1,3}$; these are in bijection with triangulations of a convex $(n-1)$-gon. The statement is presented as an open problem in the paper, and no resolution is supplied.