Exact density of radius-one type-B codes in the square grid

Determine the exact minimum density of $(1,\leq 2)$-locating-dominating codes of type B in the infinite square grid $\mathbf{G_4}$, improving the currently known bounds $\frac{2}{5}\leq D^B_{1,2}(\mathbf{G_4})\leq\frac{5}{11}$.

Background

The paper determines all minimum densities for $(1,\leq\ell)$-locating-dominating codes of types A and B in the infinite triangular grid for radius $r=1$. In its comparison with other grids, it identifies the infinite square grid as having one unresolved radius-one case: type-B codes with $\ell=2$.

The exact value of $D^B_{1,2}(\mathbf{G_4})$ is not known; the paper records a lower bound of $\frac{2}{5}$ and an upper bound of $\frac{5}{11}$ supplied by existing results. Resolving the problem would close the remaining radius-one gap for the square and king grid setting discussed in the paper.