Endowed with the operation of setwise multiplication induced by a multiplicatively written monoid $M$ on its parts, the non-empty finite subsets of $M$ containing the identity $1_M$ form themselves a monoid, herein called the reduced power monoid of $M$. We address the question of whether the reduced power monoid of a monoid $H$ is (monoid-)isomorphic to the reduced power monoid of a monoid $K$ if and only if $H$ is isomorphic to $K$. In particular, we prove a recent conjecture of P.-Y. Bienvenu and A. Geroldinger (somehow related to a classical problem of T. Tamura and J. Shafer from the late 1960s) concerning the special case where $H$ and $K$ are numerical monoids (in fact, we will treat the more general case in which $K$ is a positively orderable monoid).