The matching bound conjecture for 1-planar graphs of minimum degree 6

Let $N$ be a fixed positive integer. A 1-planar graph is a graph that admits a drawing in which each edge is crossed at most once. Consider a 1-planar graph with minimum degree at least $6$ and $n\geq N$ vertices. Matching bound conjecture. Every such graph has a matching of size at least

$$\frac{3}{7}n+O(1).$$

The preceding construction gives 1-planar graphs of minimum degree $6$ whose maximum matching has size at most $\frac{3}{7}n+\frac{4}{7}$, so the conjectured linear coefficient would be tight up to an additive constant. The source provides no resolution of the conjecture.