Monotonicity in drift p of recurrence probability for the frog model without death

Prove that, for each fixed integer d ≥ 2, the probability that the frog model with no death (q = 1) and drift parameter p toward the root on the infinite rooted d-ary tree T_d is recurrent is a monotone (nondecreasing) function of p.

Background

The drifted frog model on trees, introduced to promote recurrence by biasing moves toward the root, has been studied extensively in the case without death (q = 1). Several works have shown that sufficiently large drift can induce recurrence with positive probability.

A central conjecture in this line of research is that increasing drift should not decrease the chance of recurrence, i.e., the recurrence probability should be monotone in p. Despite progress in bounding thresholds for recurrence under drift, establishing this monotonicity remains open and is explicitly highlighted by the authors.