Nordhaus–Gaddum product lower-bound conjecture for the Hadwiger number

Let $G_1,\ldots,G_r$ be graphs on a common vertex set of size $n$, whose edge sets partition the edges of $K_n$, and let $\ngpl{\eta}(r;n)$ denote the minimum of $\prod_{i=1}^r\eta(G_i)$ over all such $r$-decompositions, where $\eta(G)$ is the Hadwiger number of $G$. Product lower-bound conjecture. For all $r\geq 2$ and $n\geq 1$,

$$\ngpl{\eta}(r;n)=n.$$

The source notes that this conjecture is implied by Hadwiger's conjecture. The paper proves only the bounds $(0.513)^{r-2}n\leq\ngpl{\eta}(r;n)\leq n$.