Exhaustiveness conjecture for factorized $A_2$-Leonard pairs

A factorized $A_2$-Leonard pair is an object defined in the paper, and Sections and give lists of examples of such pairs. Exhaustiveness conjecture. The list of factorized $A_2$-Leonard pairs given in Sections and is exhaustive. This conjecture would classify the examples of factorized $A_2$-Leonard pairs considered in the paper; the source does not state that the classification has been proved or disproved.