Asymptotic maximum conjecture for Kemeny's constant of threshold graphs

Let $n\geq 3$. For a threshold graph $G$ of order $n$, let $\mathcal{K}(G)$ denote its Kemeny constant, and let $r$ be one of the integers specified in the preceding proposition: $r\in\{\lfloor\sqrt{2n}\rfloor-1,\lfloor\sqrt{2n}\rfloor\}$ for $3\leq n\leq 20$, and $r\in\{\lfloor\sqrt{2n}\rfloor,\lfloor\sqrt{2n}\rfloor+1\}$ for $n\geq 21$. Asymptotic maximum conjecture. The maximum value of $\mathcal{K}(G)$ over all threshold graphs $G$ of order $n$ is

$$n+\frac{\sqrt{n}}{2}+\mathcal{O}(1),$$

and this maximum is achieved by the graph with construction code $0\mathbf{1}^r\mathbf{0}^{n-r-2}1$. The preceding proposition determines the maximizing values of $r$ within this family, while the global assertion over all threshold graphs and the stated asymptotic remain conjectural.