On tree decompositions whose trees are minors

In 2019, Dvořák asked whether every connected graph $G$ has a tree decomposition $(T,\mathcal{B})$ so that $T$ is a subgraph of $G$ and the width of $(T,\mathcal{B})$ is bounded by a function of the treewidth of $G$. We prove that this is false, even when $G$ has treewidth $2$ and $T$ is allowed to be a minor of $G$.