Positive drift for a bi-invariant metric on a free group

Björklund asked whether the drift of a symmetric random walk on a countable group vanishes for every bi-invariant metric with finite first moment. We show that it does not. On the free group $\mathbb{F}_r$ with $r\ge3$, let $\mu$ be the uniform measure on the $2r$ generators and their inverses, and let $L(g)$ be the least number of letters that must be deleted from a word representing $g$ so that the remaining word represents the identity. Then $d(g,h)=L(g^{-1}h)$ is a bi-invariant metric, every step has length one, and the drift of the $\mu$-walk with respect to $d$ is at least $c_r>0$, where $c_r$ is the root of an explicit entropy equation; for instance $c_{32}=1/2$. The proof is a first moment bound over noncrossing matchings.