On tree decompositions whose trees are minors

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