Desbois' roof expectation conjecture for locally free groups

Let ${\cal LF}_n^+$ be the locally free semigroup (the heap) and ${\cal LF}_n$ the locally free group (the colored heap). For a random walk word $w_N$, let $T(w_N)$ denote its roof, namely the set of achievable elements, and let $E\#T(w_N)$ be the expected size of the roof. Desbois' roof expectation conjecture. The expected roof sizes for the heap and the colored heap coincide for $n\gg 1$. Hence

$$\lim_{N\to\infty} E\#T(w_N)=\frac{n}{3}.$$

This conjecture concerns the asymptotic effect of cancellations in the group compared with the semigroup and would determine the limiting roof expectation in the large-$n$ stationary regime. The source attributes the hypothesis to J. Desbois; no resolution is given here.