Positive speed conjecture for tagged particles on Galton–Watson trees

Let a Galton–Watson tree be supercritical and condition it on survival. Consider the simple exclusion process on this tree in the constant speed model and in the variable speed model, and let the tagged particles denote distinguished particles in these processes. Positive speed conjecture. In both models, the tagged particles almost surely have a positive linear speed. This proposes extending the article's results from augmented Galton–Watson trees without leaves to Galton–Watson trees that may die out; the statement remains open in the supplied source.