Oka property of complements of closed subsets of R² in C²

Determine whether the complement \(\mathbb{C}^2 \setminus S\) is an Oka manifold for every relatively closed subset \(S \subset \mathbb{R}^2\).

Background

Kusakabe’s complement theorem, together with earlier work, had covered all pairs $(n,k)$ concerning complements of relatively closed subsets of the standard real subspaces $\mathbb{R}^k \subset \mathbb{C}^n$, except $(2,1)$, $(2,2)$, and $(3,3)$. The pair $(2,1)$ was subsequently settled, while the pairs $(2,2)$ and $(3,3)$ were recorded as open. The paper resolves the $(3,3)$ case by proving that $\mathbb{C}^3 \setminus S$ is Oka for every relatively closed $S \subset \mathbb{R}^3$, but explicitly leaves the $(2,2)$ case unaddressed.