The tree-decomposition minor conjecture

Let $G$ be a connected graph, let $T$ be a tree, and let $\mathcal{B}$ be the bags of a tree decomposition $(T,\mathcal{B})$ of $G$. The width of this decomposition is the maximum, over all $x\in V(T)$, of $|\{v\in V(G):x\in T_v\}|-1$, and $\operatorname{tw}(G)$ denotes the minimum width of a tree decomposition of $G$. A graph $T$ is a minor of $G$ if it can be obtained from $G$ by vertex and edge deletions and edge contractions.

The tree-decomposition minor conjecture. There is a function $f$ such that every connected graph $G$ has a tree decomposition $(T,\mathcal{B})$ of width at most $f(\operatorname{tw}(G))$ such that $T$ is a minor of $G$.

The paper states that this conjecture is false, even for connected graphs of treewidth $2$, and uses it as the first of three progressively stronger conjectures in the route to its main theorem.