Infinite mean offspring number in the I2 case

Let $0<\alpha\leq 1$ and consider the allometric setting, with $m_{x_0,R}(\xi_0)$ denoting the mean number of offspring from initial state $x_0$, radius parameter $R$, and initial trait $\xi_0$. Infinite-mean conjecture. If the $I_2$ condition holds, then

$$\forall x_0>0,\ \forall R\geq 0,\ \forall \xi_0>0,\qquad m_{x_0,R}(\xi_0)=+\infty.$$

This conjecture describes the claimed contrast between the $I_1$ and $I_2$ cases: finiteness is proved in the $I_1$ case for the corresponding quantity, whereas infinitude in the $I_2$ case remains conjectural.