Large graphs with bounded independence number have swap sets

Let $G$ be a graph, let $\alpha(G)$ denote its independence number, and let a swap set mean the graph structure defined in the paper that supports the relevant disjoint dominating sets and matching. Bounded-independence swap-set conjecture. For every constant $x$ there is a positive constant $y$ such that, if

$$\alpha(G)=x \quad\text{and}\quad |V(G)|\geq y,$$

then $G$ has a swap set.

The conjecture predicts that sufficiently large graphs with fixed independence number necessarily possess swap sets. The supplied text gives no resolution status.