Polynomial-time solvability of Directed Plane Strong Connectivity Augmentation

Let $D$ be a connected plane digraph and let $k$ be an integer. The Directed Plane Strong Connectivity Augmentation problem asks whether there is a set $X\subseteq V(D)^2$ with $|X|\leq k$ such that $D+X$ is strongly connected, directed, and plane for the same embedding of $D$. Directed-PSCA conjecture. The Directed Plane Strong Connectivity Augmentation problem is polynomial-time solvable. The paper establishes fixed-parameter tractability for this problem, while polynomial-time solvability remains open.