Random induced subcomplexes of thick finite buildings are asymptotically spherical

Let $d \in \mathbb{N}$ and let $(\Delta_n)_{n \in \mathbb{N}}$ be a sequence of finite $d$-dimensional buildings whose thickness satisfies

$$\operatorname{th}(\Delta_n) \rightarrow \infty$$

as $n \rightarrow \infty$. Let $\mathcal{P}(\Delta_n)$ denote the induced subcomplexes of $\Delta_n$, and let $\mathcal{S}(\Delta_n)$ denote the set of $d$-dimensional spherical subcomplexes. The general spherical-subcomplex conjecture. Then

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

This generalizes the type $A$ expectation to arbitrary sequences of finite buildings of fixed dimension and increasing thickness. It is stated as an expected consequence of the preceding conjecture, and no proof or resolution is supplied.