Concatenation-normality-large APS

Definition
An APS of groups $$(G,\Phi)$$ is termed concatenation-normality-large if for any natural numbers $$m,n$$ the block concatenation map:

$$\Phi_{m,n}:G_m \times G_n \to G_{m+n}$$

is a normality-large subgroup of $$G_{m+n}$$: in other words, its intersection with any nontrivial normal subgroup is again nontrivial.