  Here, we study the long-term behaviour of the non-explosion probability for
continuous-state branching processes in a L\'evy environment when the branching
mechanism is given by the negative of the Laplace exponent of a subordinator.
In order to do so, we study the law of this family of processes in the infinite
mean case and provide necessary and sufficient conditions for the process to be
conservative, i.e. that the process does not explode in finite time a.s. In
addition, we establish precise rates for the non-explosion probabilities in the
subcritical and critical regimes, first found by Palau et al. [19] in the case
when the branching mechanism is given by the negative of the Laplace exponent
of a stable subordinator.
