The vertex-transitive graph lower-bound conjecture for loop-erased random walk

Let $G$ be a finite vertex-transitive graph, and let $b$ and $e$ be two random vertices of $G$. Let $R$ be a random walk starting from $b$ and stopped when it hits $e$.

Loop-erased random walk lower-bound conjecture. The expected length of the loop-erasure of $R$ satisfies

$$\mathbb{E}\#\operatorname{LE}(R)\geq c\sqrt{|G|}.$$

Here $c$ is an absolute positive constant. The vertex-transitivity assumption excludes highly non-transitive examples such as a tree, where the loop-erased walk follows the unique path and can have length of order $\log N$. The conjecture proposes a universal mean-field lower bound for loop-erased random walks on finite vertex-transitive graphs.