We compute the evidence in favour of two models, one based on field-aligned thermal conduction alone and another that includes thermal misbalance as well, in explaining the damping of slow magneto-acoustic waves in hot coronal loops. Our analysis is based on the computation of the marginal likelihood and the Bayes factor for the two damping models. We quantify their merit in explaining the apparent relationship between slow mode periods and damping times, measured with SOHO/SUMER in a set of hot coronal loops. The results indicate evidence in favour of the model with thermal misbalance in the majority of the sample, with a small population of loops for which thermal conduction alone is more plausible. The apparent possibility of two different regimes of slow-wave damping, if due to differences between the loops of host active regions and/or the photospheric dynamics, may help with revealing the coronal heating mechanism.