  I show that if a finite-dimensional density matrix has strictly smaller von
Neumann entropy than a second one of the same dimension (and the rank is not
bigger), then sufficiently (but finitely) many tensor-copies of the first
density matrix majorize a density matrix whose single-body marginals are all
exactly equal to the second density matrix. This implies an affirmative
solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et
al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC
transfer to the classical setting of finite-dimensional probability vectors
(with permutations of entries instead of unitary transformations for the CEC).
