Kalai's intersecting triangulations conjecture for polygons

Let $n$ be a positive integer, and let $\mathcal F$ be a family of triangulations of the $n$-gon. The family is intersecting when every two members share a common diagonal.

Kalai's conjecture. Every intersecting family $\mathcal F$ has at most as many elements as there are triangulations of the $(n-1)$-gon.

An extremal family is obtained by fixing an ear diagonal and taking all triangulations containing it, so the bound is sharp. This is the EKR property for triangulations of a polygon and motivates the broader conjectures for flag simplicial complexes.