Weighted metric dimension on outerplanar graphs

Determine the computational complexity of the vertex-weighted Metric Dimension problem on outerplanar graphs.

Background

The unweighted Metric Dimension problem is polynomial-time solvable on outerplanar graphs. The paper notes that the vertex-weighted version is known to be solvable on classes where the unweighted problem was already known to be polynomial in several cases, while the outerplanar case was specifically posed as an unresolved question.