Unbounded gap conjecture between outerthickness and uncrossed number

Let $\theta_o(G)$ denote the outerthickness of a graph $G$, and let $\operatorname{unc}(G)$ denote its uncrossed number. Unbounded-gap conjecture. For every positive integer $k$, there exists a graph $G$ such that

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

This conjecture is attributed in the source to a prior conjecture and would follow from proving tightness of the paper's bound for a suitable constant-density graph class. Its resolution would establish that the difference between outerthickness and uncrossed number is unbounded.