Connectedness conjecture for the Hasse diagram of classical tilting modules

Let $B=\operatorname{End}_\Lambda(T_M)$, let ${\mathcal T}_B^{\rm cl}$ denote the relevant class of classical tilting $B$-modules, and let ${\mathcal F}(\Delta)$ be the category defined in the surrounding text. Connectedness conjecture. The Hasse diagram of

$$({\mathcal T}_B^{\rm cl} \cap {\mathcal F}(\Delta),\le)$$

is connected. This is a combinatorial connectedness assertion about the poset of classical tilting modules in ${\mathcal F}(\Delta)$; the source gives no evidence of a proof or counterexample.