Conjectured stable fluctuations of total length for regularly varying Lambda-coalescents

Let $g=\frac{1}{2}(\sqrt{5}+1)$ be the golden ratio. Suppose that the $\Lambda$-coalescent is regularly varying with exponent $1<\alpha<g$, meaning that

$$\int_y^1 \frac{\Lambda(dp)}{p^2}=y^{-\alpha}L(y^{-1}), \qquad 0<y<1,$$

where $L$ is slowly varying at infinity. Let $(a_n)_{n\ge1}$ satisfy $a_n\to\infty$, $a_n=o(n)$, and

$$\frac{L(n/a_n)a_n^{-\alpha}}{\Gamma(2-\alpha)L(n)}=\frac{1+o(1)}{n}.$$

Total-length fluctuation conjecture. As $n\to\infty$,

$$\frac{\ell_n-\int_2^n \frac{x}{\mu(x)}\\,dx}{a_n n^{1-\alpha}/L(n)}\stackrel{\rm d}{\longrightarrow}-c\zeta,$$

where

$$c=\frac{(\alpha-1)^{1+1/\alpha}}{(1+\alpha-\alpha^2)^{1/\alpha}\Gamma(2-\alpha)},$$

and $\zeta$ is a stable random variable with index $\alpha$ characterized by $\mathbb E[\zeta]=0$, $\mathbb P(\zeta>z)\sim z^{-\alpha}$, and $\mathbb P(\zeta<-z)=o(z^{-\alpha})$ as $z\to\infty$. The conjecture predicts a stable limit law for total-length fluctuations in the stated regularly varying regime.