3 problems

For every $Z$-group $G$ and every normal subgroup $N\trianglelefteq G$, is $N$ again a $Z$-group? Equivalently, is the class of all $Z$-groups closed under taking normal subgroups?

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

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