Uncrossed number can differ arbitrarily from outerthickness

Let ${\mathrm{unc}}(G)$ denote the uncrossed number of a graph $G$, and let $\theta_o(G)$ denote its outerthickness. For every positive integer $k$, there is a graph $G$ such that

$$\theta_o(G)-{\mathrm{unc}}(G) \geq k.$$

Uncrossed-number separation conjecture. The uncrossed number can be arbitrarily far apart from the outerthickness. This conjecture asks whether the difference between these two graph parameters is unbounded; for the complete and complete bipartite graphs studied in the paper, their difference is never larger than one, so those classes do not establish the conjectured separation.