Finite nuclear images of conuclear images of negative cones

Let an $\ell$-group be a lattice-ordered group, let a negative cone be its submonoid of elements below the identity, and let a conuclear image be an image obtained from a conucleus. A residuated lattice is called integral when its identity is its greatest element. Consider finite nuclear images of conuclear images of negative cones of $\ell$-groups.

Finite nuclear-image conjecture. The finite nuclear images of conuclear images of negative cones of $\ell$-groups are precisely the finite integral residuated lattices.

The conjecture proposes replacing cancellative residuated lattices by conuclear images of $\ell$-groups in the corresponding finite representation theorem. The paper records this as an open problem whose validity is not known.