Conjectured pure-jump process limit for total length of Beta-coalescents

Let $\ell_n(t)$ be the tree length of the coalescent tree $\left(\mathcal T_n(t)\right)_{t\in\mathbb R}$. Let $g=\frac{1}{2}(1+\sqrt{5})$. For Beta$(2-\alpha,\alpha)$-coalescents with $g<\alpha<2$, define

$$c=\frac{\Gamma(\alpha)\alpha(\alpha-1)}{2-\alpha}.$$

Total-length process conjecture. The sequence of stationary processes

$$\left(\ell_n(t)-cn^{2-\alpha}\right)_{-\infty<t<\infty}$$

converges in finite-dimensional distribution to a stationary pure-jump process. This conjecture extends the proposed fluctuation picture from the Kingman and lower-$\alpha$ regimes to Beta-coalescents with $g<\alpha<2$.