Metric dimension on fixed k-outerplanar graphs and grids with holes

Determine the computational complexity of Metric Dimension on $k$-outerplanar graphs for each fixed $k\ge2$ and on grids with holes, and determine whether the constructed graphs $G_\varphi$ are $k$-outerplanar for any fixed $k$.

Background

Prior work cited in the paper asks for the complexity of Metric Dimension on fixed-$k$ outerplanar graphs for $k\ge2$ and on grids with holes. The paper’s hardness construction would resolve the fixed-$k$ outerplanar question if its graphs belonged to such a class, but the authors do not establish that structural property.