Conjecture on total internal branch length in dust-free coalescents

Let $\hat \ell_n:=\ell_n-\bar \ell_n$ be the total internal branch length of an $n$-coalescent, where $\ell_n$ is its total length and $\bar \ell_n$ is its total external branch length. For a dust-free $\Lambda$-coalescent, as $n\to\infty$,

$$\hat \ell_n \stackrel P \sim \int_2^n \left(\frac{x}{\mu(x)}-\frac{n}{\mu(n)}\right)\\,dx.$$

The analogous result is known for a class of coalescents containing the Bolthausen–Sznitman coalescent, and the conjecture proposes that the asymptotic law holds for all dust-free $\Lambda$-coalescents.