Random induced subcomplexes of type A buildings are asymptotically spherical

Let $\Delta_{k,n}$ be the finite spherical building of type $A_k$, let $\mathcal{P}(\Delta_{k,n})$ denote its induced subcomplexes, and let $\mathcal{S}(\Delta_{k,n})$ denote the set of induced subcomplexes that are spherical and have the same dimension as $\Delta_{k,n}$. The spherical-subcomplex conjecture. For every natural number $k$,

$$\lim_{n \rightarrow \infty} \frac{\left\lvert \mathcal{S}(\Delta_{k,n}) \right\rvert}{\left\lvert \mathcal{P}(\Delta_{k,n}) \right\rvert}=1.$$

This asserts that almost every induced subcomplex is a top-dimensional spherical complex, strengthening the preceding result that almost every induced subcomplex has nontrivial top-dimensional reduced homology. The conjecture is presented as an expected statement and no resolution is given.