The conjectured bases for distinguished subspaces of the fundamental module

Consider the $Q$-polynomial distance-regular graph $\Gamma=(X,\mathcal R)$ of diameter $D$, its standard module $V$, and the fundamental $\mathbb T$-submodule $\Lambda\subseteq V^{\otimes 3}$. Let $E_r^{*(k)}$ and $E_r^{(k)}$ denote the dual and ordinary primitive-idempotent operators acting on the $k$th tensor factor, and let $P_{a,b,c}$ and $Q_{a,b,c}$ be the vectors defined in the paper. The conjectured bases for distinguished subspaces of the fundamental module. The following sets are conjectured to be bases of the indicated subspaces:

$$\begin{array}{c|ccc} \text{Subspace}&E_0^{*(1)}\Lambda&E_0^{*(2)}\Lambda&E_0^{*(3)}\Lambda\\ \hline \text{Conjectured basis}&\{P_{0,i,i}\}_{i=0}^D&\{P_{i,0,i}\}_{i=0}^D&\{P_{i,i,0}\}_{i=0}^D \end{array}$$ $$\begin{array}{c|ccc} \text{Subspace}&E_0^{(1)}\Lambda&E_0^{(2)}\Lambda&E_0^{(3)}\Lambda\\ \hline \text{Conjectured basis}&\{Q_{0,i,i}\}_{i=0}^D&\{Q_{i,0,i}\}_{i=0}^D&\{Q_{i,i,0}\}_{i=0}^D \end{array}$$

These conjectured bases would give explicit descriptions of six natural subspaces associated with the three tensor factors and the ordinary or dual idempotents. Their verification is posed as a direction for future research.