The endpoint-rooted path conjecture for cover cost-to-time ratio

Let $G$ be a rooted graph on $n$ vertices, with root $r$, and let $cc_r(G)$ and $CT_r(G)$ denote its cover cost and cover time from $r$. The path on $n$ vertices rooted at an endpoint maximises the ratio

$$\frac{cc_r(G)}{CT_r(G)}$$

over all rooted graphs $G$ on $n$ vertices.

This conjecture concerns extremal comparisons between cover cost and cover time, which can have substantially different orders of magnitude depending on the graph. A proof would provide an extremal bound for using cover cost to estimate cover time; the source gives no resolution.