Multiplicity-one conjecture for common eigenspaces of alternating elements

Let $\Gamma$ be the distance-regular graph and let $\mathbb T$ be its subconstituent algebra as in Section 11. For an irreducible $\mathbb T$-module $V$, consider the common eigenspaces for each of the families

$$\lbrace W_{-k}\rbrace_{k \in \mathbb N},\qquad \lbrace W_{k+1}\rbrace_{k \in \mathbb N},\qquad \lbrace G_{k+1}\rbrace_{k \in \mathbb N},\qquad \lbrace {\tilde G}_{k+1}\rbrace_{k \in \mathbb N}.$$

Multiplicity-one conjecture. For every irreducible $\mathbb T$-module, the common eigenspaces for each of these four families all have dimension one. This conjecture would give a uniform multiplicity-one description of the four natural eigen decompositions associated with the alternating elements of the subconstituent algebra. The surrounding results establish simultaneous eigenspace decompositions for some of these families, but the asserted one-dimensionality is presented as a direction for further research.