Improper conflict-free and unique-maximum coloring conjecture for planar and outerplanar graphs

Let $\mathcal{P}$ be the class of planar graphs and $\mathcal{O}$ the class of outerplanar graphs. Write $\chi_{\mathrm{iCFo}}$, $\chi_{\mathrm{iCFc}}$, $\chi_{\mathrm{iUMo}}$, and $\chi_{\mathrm{iUMc}}$ for the corresponding improper conflict-free and improper unique-maximum chromatic numbers, with respect to open or closed neighborhoods as indicated by the subscripts.

Improper coloring conjecture. For the classes $\mathcal{P}$ and $\mathcal{O}$, the following equalities hold:

$$\chi_{\mathrm{iCFo}}(\mathcal{P})=4,\quad \chi_{\mathrm{iCFc}}(\mathcal{P})=3,\quad \chi_{\mathrm{iUMo}}(\mathcal{P})=5,\quad \chi_{\mathrm{iUMc}}(\mathcal{P})=4,$$ $$\chi_{\mathrm{iCFo}}(\mathcal{O})=3,\quad \chi_{\mathrm{iUMo}}(\mathcal{O})=4,\quad \chi_{\mathrm{iUMc}}(\mathcal{O})=3.$$

The conjecture proposes that the currently known lower bounds are tight for all listed improper coloring parameters; the corresponding gaps between lower and upper bounds are unresolved.