The pairwise-Leonard criterion for Leonard triples

Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*,A^\varepsilon:V\to V$ be linear transformations. A Leonard pair is a pair of transformations for which one is diagonal and the other is irreducible tridiagonal in suitable bases. The pairwise-Leonard criterion. If each two-element subset of $A,A^*,A^\varepsilon$ is a Leonard pair on $V$, then $A,A^*,A^\varepsilon$ is a Leonard triple on $V$, meaning that each member can be diagonalized in a basis in which the other two are irreducible tridiagonal. The problem asks for a classification of Leonard triples; no resolution of this implication is supplied in the source.