Matching the logarithmic lower bound on arbitrarily weighted outerplanar graphs

Determine whether any strategy for the k-Canadian Traveller Problem on arbitrarily weighted outerplanar graphs can achieve a competitive ratio matching the known lower bound of Ω(log k).

Background

The paper establishes an asymptotically tight O(√k) competitive ratio for deterministic strategies on unit-weighted solid grid graphs. In its conclusion, it contrasts this result with prior work showing a lower bound of Ω(log k) for arbitrarily weighted outerplanar graphs. The authors explicitly identify as unresolved whether this lower bound is achievable by some strategy, which would characterize the optimal competitive ratio in that setting.