Star arboricity relaxed book thickness of $K_n$

A book embedding of the complete graph $K_n$ needs $\lceil \frac{n}{2}\rceil$ pages and the page-subgraphs can be chosen to be spanning paths (for $n$ even) and one spanning star for $n$ odd. We show that all page-subgraphs can be chosen to be {\rm star forests} by including one extra {\rm cross-cap} page or two new ordinary pages.