Asymptotic codimension-one Cheeger constant conjecture for finite buildings

Let $A_{n+1}({\mathbb F}_q)$ be the $n$-dimensional building associated with the projective geometry over the finite field ${\mathbb F}_q$, and let $h_{n-1}$ denote its normalized codimension-one Cheeger constant. Asymptotic Cheeger constant conjecture. For fixed $n$ and $q\to\infty$, one has

$$h_{n-1}\left(A_{n+1}({\mathbb F}_q)\right)=1-o(1).$$

This would improve the bounds for the normalized Cheeger constants in the paper, extending the known one-dimensional result for the points-versus-lines graph of the Desarguesian projective plane. The source presents this as a likely improvement rather than reporting a resolution.