Near-perfect matching conjecture for 5-connected 1-planar graphs

Let $G$ be a 5-connected 1-planar graph with $n$ vertices. A matching is a set of pairwise vertex-disjoint edges.

Near-perfect matching conjecture. Every 5-connected 1-planar graph has a matching of size

$$\frac{n}{2}-O(1).$$

The claim is motivated by a construction of 5-connected 1-planar graphs whose independent stellation-vertices force at least two unmatched vertices in every matching, while the graphs otherwise have near-perfect matchings. Whether higher connectivity generally implies Hamiltonicity in all 1-planar graphs remains an open question.