Star-like graph mode conjecture for domination polynomials

For a graph $G$, let $D(G,x)$ be its domination polynomial, and call the index of a largest coefficient its mode. A graph is star-like in the sense intended by the source. Star-like graph mode conjecture. For each order $n$, there is a star-like graph whose domination polynomial has its mode at the largest index among all graphs of order $n$ without isolated vertices.

The conjecture is motivated by the relationship between the mode of a domination polynomial and the average order of dominating sets, together with computational evidence. The supplied material gives no resolution, so the conjecture remains open.