Orthogonality conjecture for the four split decompositions

Under Assumption, let $D$ be the diameter and let ${\tilde V}^{\downarrow \downarrow}_{ij}$, ${\tilde V}^{\uparrow \uparrow}_{rs}$, ${\tilde V}^{\downarrow \uparrow}_{ij}$, and ${\tilde V}^{\uparrow \downarrow}_{rs}$ denote the four split-decomposition spaces. Orthogonality conjecture. For $0 \leq i,j\leq D$, the spaces

$${\tilde V}^{\downarrow \downarrow}_{ij}\quad\text{and}\quad {\tilde V}^{\uparrow \uparrow}_{rs}$$

are orthogonal unless $i+r=D$ and $j+s=D$. Moreover,

$${\tilde V}^{\downarrow \uparrow}_{ij}\quad\text{and}\quad {\tilde V}^{\uparrow \downarrow}_{rs}$$

are orthogonal unless $i+r=D$ and $j+s=D$. This would describe the orthogonality pattern among the split-decomposition spaces under the stated assumption; the source gives no resolution of the conjecture.