The multitype $b[\Lambda]b[0m$-coalescent differential-equation speed conjecture

Let $N_i(t)$ denote the number of blocks of type $i$ at time $t$, and let $b[\Phi]b[0m:\mathbb{R}^d\to\mathbb{R}^d$ have components

$$\Phi_i(n_1,\ldots,n_d):=\sum_{j\ne i}(\rho_{j\to i}n_j-\rho_{i\to j}n_i)-\rho_{ii\to i}\frac{n_i(n_i-1)}{2}+\sum_{j=1}^d\int_{[0,1]^d}\delta_{i,j}\left\{1-\prod_{k=1}^d(1-s_k)^{n_k}\right\}-s_i n_i\,Q_{\to j}(\mathrm{d}s).$$

Multitype coalescent speed conjecture. Provided the conditions of the theorem establishing coming down from infinity hold, the ordinary differential equation

$$\dot{v}(t)=\Phi(v(t)),\qquad v_i(0)=+\infty$$

has a unique solution such that, when the process starts with infinitely many blocks of each type,

$$\lim_{t\downarrow0}\frac{N_i(t)}{v_i(t)}=1$$

almost surely for each type $i$. The paper gives a conclusive criterion for whether a multitype $b[\Lambda]b[0m$-coalescent comes down from infinity, but states that the speed of coming down is more delicate and remains unclear. This conjecture proposes that the speed is described by the deterministic flow generated by the expected block-count drift, extending the known single-type result of Berestycki, Berestycki and Limic.