3 problems

Toroidal shallow-minor conjecture. There is no constant $k\in\mathbb N$ with the following property: every toroidal graph is a congestion-$k$ depth-$k$ minor of a planar graph.

Clique-separator conjecture. There is a clique cover $C$ in $G$ such that removing

Shallow-minor star conjecture. If $G$ does not have an induced star on $s$ leaves, then, for every $t\ge 0$,