Nuclear images of sub-sℓ-monoids of negative cones

Let an s$\ell$-monoid be a partially ordered monoid equipped with the relevant semilattice operations, and let an $\ell$-group be a lattice-ordered group. A negative cone is the submonoid of elements below the identity, and a nuclear image is an image obtained from a nucleus. Consider sub-s$\ell$-monoids of negative cones of $\ell$-groups.

Nuclear-image conjecture. The nuclear images of sub-s$\ell$-monoids of negative cones of $\ell$-groups are precisely the integral s$\ell$-monoids.

This would follow if the proof-theoretic argument used for pomonoids could be adapted to s$\ell$-monoids, closing the gap between cancellative structures and submonoids of groups. The claim remains open beyond the integral and commutative cases.