Congested shallow-minor obstruction for toroidal graphs

A graph $H$ is a congestion-$c$ depth-$d$ minor of a graph $G$ if it has a congestion-$c$ depth-$d$ minor model in $G$: each vertex of $H$ is represented by a connected subgraph of $G$ of radius at most $d$, adjacent vertices of $H$ have intersecting or adjacent branch sets, and every vertex of $G$ belongs to at most $c$ branch sets.

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.

This is proposed as a combinatorial stepping stone toward the surface transduction-order conjecture. The survey presents it as an open question.