Unbounded ratio between path and distance parameters

Determine whether, for every positive integer k, there exists a graph G and a vertex v in V(G) such that the ratio f_p(G,v)/f_d(G,v) exceeds k.

Background

The paper compares the classical distance-variant parameter f_d(G,v) with the path-variant parameter f_p(G,v) of the Explorer–Director game. For the constructed cuttlefish graphs CF_n, the authors prove that f_p(CF_n,v)−f_d(CF_n,v) can exceed any fixed additive constant as n grows, establishing an arbitrarily large additive separation.

The unresolved question asks whether this separation can also be made arbitrarily large multiplicatively: specifically, whether the path-variant value can be more than k times the distance-variant value for some graph and starting vertex, for every prescribed k.