Sphericality conjecture for upper intervals in partial decomposition posets of buildings

Let $\Delta$ be a spherical building, and let $\sigma\in\Delta$ be any non-empty simplex. Let ${\mathcal{OPD}}(\Delta)$ and ${\mathcal{PD}}(\Delta)$ denote the ordered partial decomposition and partial decomposition posets, respectively. Upper-interval sphericality conjecture. The upper intervals satisfy

$${\mathcal{OPD}}(\Delta)_{\succ \sigma}\text{ and }{\mathcal{PD}}(\Delta)_{\succ \sigma}\text{ are spherical of dimension }2\dim\Delta-\dim\sigma.$$

In particular, ${\mathcal{OPD}}(\Delta)$ and ${\mathcal{PD}}(\Delta)$ should be Cohen–Macaulay of dimension $2\dim\Delta+1$. The preceding results establish related sphericality statements, but these upper intervals are described as harder to control and remain conjectural in the supplied text.