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On a conjectured property of the von Neumann entropy valid in the commutative case

It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian $H$ of the system satisfies the condition $\Tr\exp(-cH)<+\infty$ for any $c>0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian $H$ satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.