The pagenumber conjecture for complete expansion graphs

Let $K_n$ be the complete graph on $n$ vertices, let $\mathcal{E}_{c}(K_n)$ denote its complete expansion graph, and let $pn(G)$ be the pagenumber of a graph $G$. For a positive integer $m$, set $n=2m+1$. The pagenumber conjecture.

$$pn(\mathcal{E}_{c}(K_n))=\left\lceil\frac{n}{2}\right\rceil.$$

The preceding theorem gives lower and upper bounds differing by one for odd $n$; the equality is known when $m=2$, equivalently for $\mathcal{E}_{c}(K_5)$, but the general case remains open.