The Nordhaus-Gaddum upper-bound conjecture for the Cheeger constant

Let $G$ be a graph, let $G^c$ be its complement, and let $h(G)$ denote the Cheeger constant of $G$. Write $S_n$ for the star graph on $n$ vertices, and let $K_3\vee E_{n-3}$ denote the join of the complete graph on three vertices and the edgeless graph on $n-3$ vertices.

Cheeger-constant Nordhaus-Gaddum conjecture. If $G\neq S_n$, then

$$\max\{h(G),h(G^c)\}\leq h(K_3\vee E_{n-3}).$$

The conjecture is motivated by computational and structural comparisons among graph families, but the supplied text gives no resolution or status evidence beyond presenting it as a conjecture.