Let $(m_i, i \in \mathbb{N})$ be positive weights with $\sum_{i \in \mathbb{N}} m_i^2 < 0.1$. Consider a subcritical branching process in discrete time and continuous space, started from some initial mass $x>0$, and with branching mechanism as follows: given mass $m$ in generation $n$, generation $n+1$ has total mass $$ \sum_{i \in \mathbb{N}} m_i N_i, $$ where the $N_i$ are independent and $N_i$ has distribution $\mathrm{Poisson}(m\cdot m_i)$. (One can think of a total number $N_i$ of copies of mass $m_i$ in generation $n+1$.)

Let $Z_n$ be the total mass at level $n$. Does $\sum_{n=0}^{\infty} Z_n^{1/2}$ then have exponential tails?