Commutativity conjecture for Hecke correspondences on Rapoport–Zink spaces

Let $t\neq t'$ and let $A$ be any closed formal subscheme of the Rapoport–Zink space $\mathcal{N}_n$. The correspondence $\mathcal{T}_n^{\leq t}$ is obtained by composing the correspondences $\mathcal{N}_n^{[0,t]}$ and $\mathcal{N}_n^{[t,0]}$, and $\mathbb{T}_t^A$ is the induced map on K-groups. Commutativity conjecture. The closed formal subschemes obtained by applying the two correspondences in opposite orders agree up to a locally nilpotent ideal sheaf,

$$|\mathcal{T}^{\leq t}(\mathcal{T}^{\leq t'}(A))|=|\mathcal{T}^{\leq t'}(\mathcal{T}^{\leq t}(A))|,$$

and the induced maps

$$\mathbb{T}^{\mathcal{T}^{\leq t'}(A)}_t\circ\mathbb{T}^A_{t'} \quad\text{and}\quad \mathbb{T}^{\mathcal{T}^{\leq t}(A)}_{t'}\circ\mathbb{T}^A_t$$

are identical modulo torsion. The conjecture expresses commutativity of the Hecke correspondences and is empty for $n=2$ and $n=3$.