The proof of the second convergence of Theorem 2.1 provided in the original paper relies on properties of the so called Q-process, which is not, in general, defined on the whole state space. Hence the inequality is only proved for some restricted sets of functions and measures. This weakness can be easily repaired and we provide here a proof of the inequality that applies in full generality.
"Erratum: Practical criteria for $R$-positive recurrence of unbounded semigroups." Electron. Commun. Probab. 25 1 - 2, 2020. https://doi.org/10.1214/20-ECP307