The biased edge-reinforced random walk transience conjecture

Let $\lambda>0$, and consider the $\lambda^\ast$-biased edge-reinforced random walk on $\mathbb{Z}$, in which the transition probability at a node is weighted by the current edge weights with bias parameter $\lambda$. A walk on $\mathbb{Z}$ is transient if each node is visited only finitely often almost surely.

Biased edge-reinforced random walk transience conjecture. The $\lambda^\ast$-biased edge-reinforced random walk is transient whenever $\lambda\neq1$.

The conjecture is motivated by the expected limiting effect of the bias at nodes visited infinitely often. The source states that this has not been proved; the case $\lambda=1$ is excluded because it is the unbiased model.