Equivalence of the two-sided dictionary and merger orders on Lusztig data

Let $A(\varnu)$ be the set of Lusztig data equipped with the two-sided dictionary partial order from the cited definition and the merger partial order from the cited definition. Equivalence conjecture. The partial order from the two-sided dictionary definition and the partial order from the merger definition are equivalent.

The claim compares two poset structures on Lusztig data. The supplied text gives no resolution status or further context about what the equivalence entails.