  In 2019, Dvo\v{r}\'{a}k 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 $G$.
