The radial tree-width conjecture for graphs excluding quasi-geodesic subdivisions of the triangle

Let $G$ be a connected graph. For $c,k\in\mathbb{N}$, a $c$-quasi-geodesic $(\geqslant ck)$-subdivision of $K^3$ means a subdivision of the triangle whose subdivided paths are $c$-quasi-geodesic and have length at least $ck$. The radial tree-width of $G$ is the tree-width notion defined for radial decompositions.

Radial tree-width conjecture. There is a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that, if a connected graph $G$ does not contain a $c$-quasi-geodesic $(\geqslant ck)$-subdivision of $K^3$ for some $c,k\in\mathbb{N}$, then

$$\operatorname{radial\ tree\text{-}width}(G)\leqslant f(k).$$

The preceding discussion explains that the subdivided-star argument does not directly produce a quasi-geodesic structural subgraph and presents this assertion as an unresolved question for trees. No resolution is given in the supplied text.