General sampling formula conjecture for Lambda-coalescents

Let $\Lambda$ be a finite measure on $[0,1]$ with $\Lambda(\{1\})=0$, and let $\psi(q)=\int_{[0,1]}(e^{-qx}-1+qx)x^{-2}\Lambda(dx)$. For a sample of size $n$, let $X_n$ denote either the number $A_n$ of allelic families or the number $S_n$ of segregating sites. General sampling formula conjecture. For every $\Lambda$-coalescent, the convergence

$$\frac{X_n}{\displaystyle\int_1^n q\psi(q)^{-1}\,dq}\longrightarrow\theta$$

holds in the $L^1$ sense as $n\to\infty$.