= Solution
After leaving zero, the jump chain avoids every future reset with probability
$$
q=\prod_{i=1}^{\infty}\frac1{1+\rho_i}.
$$
This probability is positive exactly when $\prod_{i\geq1}(1+\rho_i)<\infty$. If $q>0$, the probability of ever returning to zero is less than one, so the irreducible chain is transient. If $q=0$, a reset to zero occurs almost surely after each departure; by the <Strong Markov property> this happens repeatedly, so the chain is recurrent. Therefore
$$
\boxed{X\text{ is transient}\iff\prod_{i=1}^{\infty}(1+\rho_i)<\infty.}
$$
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