Quasirandom forcing in regular tournaments

A sequence of tournaments is said to be quasirandom if it behaves as a  sequence of random tournaments would. In 1991, Chung, Graham, and Wilson provided a list of equivalent properties that any sequence of random tournaments satisfies with high probability. We say that a tournament H  forces quasirandomness if every sequence that asymptotically has the expected number of copies of H is quasirandom. Nevertheless, it was shown that almost only transitive tournaments are quasirandom forcing. We then modify the definition of quasirandom forcing by considering only sequences of nearly regular tournaments and characterize all tournaments on at most five vertices that force quasirandomness in this new setting.