The limiting-hyperbola conjecture for domination roots of generalized friendship graphs

Let $F_{4,n}$ be the generalized friendship graph with parameter $n$, and consider the complex zeros of its domination polynomial $D(F_{4,n},x)$ as $n$ varies. Limiting-hyperbola conjecture. The limit of domination roots of $F_{4,n}$ is a hyperbola. The source introduces this as a question suggested by a plot of the complex domination roots, but gives no proof or precise definition of the limiting set.