The treewidth packing conjecture with a logarithmic bound

For a graph $G$, let $r,k\geq 1$ be integers, and let $f:\mathbb{N}\to\mathbb{N}$ be a function.

Treewidth packing conjecture. There is a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G$ with

$$\operatorname{tw}(G)\geq f(r)\,k\log(k+1)$$

has $k$ vertex-disjoint subgraphs $G_1,\dots,G_k$, each satisfying

$$\operatorname{tw}(G_i)\geq r.$$

This would improve the logarithmic exponent in the polynomial Grid Theorem and is suggested as a possible route toward the tight Erdős–Pósa bound for planar-minor models. The source presents it as a natural side conjecture and gives no resolution.