Recurrence conjecture for the critical biased random walk on self-avoiding-walk trees

Let $T_{\mathbb H}$ and $T_{\mathbb Z^2}$ be the trees of self-avoiding walks in the half-plane and in $\mathbb Z^2$, respectively, and let $RW_{\lambda_c}$ denote the biased random walk at the critical bias parameter $\lambda_c$.

Critical-walk recurrence conjecture. The biased random walk $RW_{\lambda_c}$ on $T_{\mathbb H}$ or $T_{\mathbb Z^2}$ is recurrent.

The conjecture is motivated by a cutset criterion and the predicted growth estimates for the number of self-avoiding walks. The supplied text gives no resolution, so recurrence at criticality remains open here.