The cardinality of Kiselman's semigroups grows double-exponentially

Let $K_n$ denote Ganyushkin-Kudryavtseva-Mazorchuk’s generalization of Kiselman’s semigroups. We show that the sequence $2^{-n\mathrm{/}2}\cdot \log \mathrm{\mid }{K}_n\mathrm{\mid }$ admits finite limits as $n$ grows to infinity both on odd and even values.