Barma–Dhar critical-bias conjecture for biased random walk on percolation clusters

Let $p\in(p_c,1)$, where $p_c=\frac{1}{2}$ is the critical probability for bond percolation on $\mathbb{Z}^2$. For the biased random walk on the infinite percolation cluster, let $\beta_\ell$ and $\beta_u$ be the constants from Theorem 1, so that positive speed holds when $1<\beta<\beta_\ell$ and zero speed holds when $\beta>\beta_u$. Define $\beta_{\rm crit}:=\beta_\ell=\beta_u$. Barma–Dhar's critical-bias conjecture. The statements of Theorem 1 hold with $\beta_{\rm crit}:=\beta_\ell=\beta_u$. This conjecture asserts that the transition from positive to zero speed occurs at a single critical bias. The paper proves only the existence of possibly distinct lower and upper thresholds, so equality remains open here.