I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth is as high as possible. Formally my problem is the following: having chosen a degree bound $d \in \mathbb{N}$, what is the "best" function $f : \mathbb{N} \to \mathbb{N}$ such that, in any graph $G$ with treewidth $k$, I can find (hopefully efficiently) a subgraph $H$ of $G$ with maximal degree $\leq d$ and treewidth $f(k)$.
Obviously we should take $d \geq 3$ as there are no high treewidth graphs with maximal degree $<3$. For $d = 3$ I know that you can take $f$ such that $f(k) = \Omega(k^{1/100})$ or so, by appealing to Chekuri and Chuzhoy's grid minor extraction result (and using it to extract a high-treewidth degree-3 graph, e.g., a wall, as a topological minor), with the computation of the subgraph being feasible (in RP). However, this is a very powerful result with an elaborate proof, so it feels wrong to use it for what looks like a much simpler problem: I would just like to find any constant-degree, high-treewidth subgraph, not a specific one like in their result. Further, the bound on $f$ is not as good as I would have hoped. Sure, it is known that it can be made $\Omega(k^{1/20})$ (up to giving up efficiency of the computation), but I would hope for something like $\Omega(k)$. So, is it possible to show that, given a graph $G$ of treewidth $k$, there is a subgraph of $G$ with constant degree and linear treewidth in $k$?
I'm also interested in the exact same question for pathwidth rather than treewidth. For pathwidth I don't know any analogue to grid minor extraction, so the problem seems even more mysterious...