Non-explosion conjecture for the interacting particle process

Let $H_r$ be the state space of height configurations and let $\widetilde X^{(z,z')}_r$ be the Markov process on $H_r$ whose deterministic component evolves each height according to $v(y)=y(y+1)$ until it reaches $r$, while its stochastic component causes the drops specified by the point configuration and neighboring heights. Non-explosion conjecture. The process $\widetilde X^{(z,z')}_r$ on $H_r$ does not explode and so has infinite life time almost surely. The claim concerns the global well-definedness of this interacting piecewise deterministic Markov process; the supplied text presents it as plausible but gives no proof or resolution.