Do Tsallis distributions really originate from the finite baths?

It is often stated that heat baths with finite degrees of freedom i.e., finite baths, are sources of Tsallis distributions for the classical Hamiltonian systems. By using well-known fundamental statistical mechanical expressions, we rigorously show that Tsallis distributions with fat tails are possible \textit{only} for finite baths with constant negative heat capacity while constant positive heat capacity finite baths yield decays with sharp cut-off with no fat tails. However, the correspondence between Tsallis distributions and finite baths holds at the expense of violating equipartition theorem for finite classical systems at equilibrium. Finally, we comment on the implications of the finite bath for the recent attempts towards a $q$-generalized central limit theorem.