The c-pathwidth lower-bound conjecture

Let $c$ be a positive integer. For a graph $G$, write $c$-$pw(G)$ for its $c$-pathwidth, $tw(G)$ for its treewidth, and $n$ for the number of vertices of $G$. The c-pathwidth lower-bound conjecture. There is a constant $\alpha_c$ and a class $\mathbf{G}_c$ of graphs of unbounded treewidth such that, for every $G\in\mathbf{G}_c$,

$$c\text{-}pw(G)\geq \alpha_c\,tw(G)\log n.$$

The source presents this as an extension of the lower bound in Proposition from ordinary pathwidth to $c$-pathwidth. Whether $c$-pathwidth can be bounded above by a function of treewidth alone is stated to be unknown, and the conjectured lower bound would show a strong separation on a class of graphs with unbounded treewidth.