The spanning-tree decomposition conjecture

Let $G$ be a connected graph, let $T$ be a spanning tree of $G$, and let $(T,\mathcal{B})$ be a tree decomposition of $G$ with width measured as the maximum bag size minus one. Write $\operatorname{tw}(G)$ for the treewidth of $G$.

The spanning-tree decomposition 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 spanning tree of $G$.

This is presented as a strengthening of the minor version and is one of three conjectures reduced successively to the next. The supplied text does not state its independent resolution, although the paper's overall strategy ultimately disproves the final, stronger conjecture.