Fix a positive integer $n$ and let $H$ be a submonoid of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_H\}$ is contained in an open half-space (where $0_H$ is the identity of $H$, i.e., the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is whether

$H$ is atomic only if it is BF-atomic.

Let me recall that a multiplicatively written monoid is atomic if every non-unit is a product of atoms (namely, non-units that do not factor as a product of non-units); and is BF-atomic if each non-unit $x$ has at least one factorization into atoms and there is an upper bound (depending on $x$) on the length of these factorizations.