Antal and Krapivsky's growth conjecture for multistage epidemics

Assume that $\varepsilon_{n,k}=\delta_{n,k}=0$. For $k=1,\ldots,K$, let $N_{n,k}$ be the number of individuals who are ever of type $k$ during the epidemic, which starts from

$$a_{n,1}(0)=1,\qquad a_{n,k}(0)=0\text{ for }k=2,\ldots,K+1.$$

Define

$$\lambda_K=\frac{2^K-1}{(K+1)2^K-1}.$$

Antal and Krapivsky's conjecture. For each $k=1,\ldots,K$, the expected number of individuals of type $k$ satisfies

$$\mathbb{E}(N_{n,k})\text{ grows like }n^{klambda_K}\text{ as }nto+\infty.$$

This conjecture, formulated in the cited work, concerns the polynomial growth of the expected numbers of individuals reaching each stage in a critical multistage epidemic. The supplied material does not state whether it has been proved or disproved.