Universal TCS characterization of containment for non-binary tree-child networks

Let $N$ and $N'$ be non-binary tree-child networks on the same leaf set. A tree-child sequence (TCS) of $N$ is a sequence of reductions associated with the tree-child network. Universal TCS containment conjecture. The network $N'$ is contained in $N$ if and only if every TCS of $N$ reduces $N'$; equivalently,

$$N'\text{ is contained in }N \quad\Longleftrightarrow\quad \text{all TCSs of }N\text{ reduce }N'.$$

The analogous statement with existence of a suitable TCS is known for the tree-child setting in the reconstructible classes, while the universal condition is proposed here for general non-binary tree-child networks.