The locally free group drift conjecture

Let ${\cal LF}_n$ be a locally free group, and consider its random walk with word length $K$ and drift $l$, the limiting expected normalized word length. Let $T(w)$ be the roof of a word $w$, and let $E\#T(w)$ be its expected size. Locally free group drift conjecture. For $n\gg 1$, the expected drift is

$$l=\frac{2}{3}.$$

The preceding calculation gives the drift as $l=(2-\alpha)/(3-\alpha)$, where $\alpha$ measures the asymmetry between roof-increasing and roof-decreasing cancellations; the conjectured value follows when the two conditional effects balance. No resolution is given in the source.