The strong 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 $G$. For each vertex $v\in V(G)$, define $T_v$ to be the subtree of $T$ induced by the vertices $x$ whose bags contain $v$. The width is the maximum, over $x\in V(T)$, of $|\{v\in V(G):x\in T_v\}|-1$, and $\operatorname{tw}(G)$ is the minimum width of a tree decomposition of $G$.

The strong 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 $G$ and, for every vertex $v$ of $G$, we have $v\in T_v$.

The condition $v\in T_v$ requires each graph vertex to belong to the bag indexed by its corresponding vertex of the spanning tree. The supplied context says that the paper reduces this conjecture to a final conjecture and then disproves that final conjecture, but it does not explicitly state the independent status of this intermediate claim.