Fix a positive integer $n$ and let $H$ be a submonoid of the (additive) monoid of integer points of a [polyhedral cone][1] of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_H\}$ is contained in an [_open_ half-space][2] (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][3] 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 has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations. 


  [1]: https://en.wikipedia.org/wiki/Convex_cone#Polyhedral_and_finitely_generated_cones
  [2]: https://en.wikipedia.org/wiki/Convex_cone#Half-spaces
  [3]: https://mathoverflow.net/questions/428351/characterizing-atomicity-in-a-commutative-domain