On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any *finite* simply connected complex should [have the same weak homotopy type as] the classifying space of a *finite* monoid." The paper was published in 1984. Has any progress been made in the last 30 years that would allow us to replace "it seems likely" by "it is known"?

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discretemonoid". $\endgroup$ – BS. Mar 10 '16 at 16:52