Is the acyclic chromatic number bounded in terms of the book thickness?

I believe that "book thickness bounds the acyclic chromatic number" was established in this paper:

Dujmovic, Vida, Attila Pór, and David R. Wood. "Track layouts of graphs." Discrete Mathematics and Theoretical Computer Science 6, no. 2 (2004). arXiv abs.

In the Abstract they say,

"As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number."

And then later (Section 5), they say,

"Note that stack-number is also called page-number and book-thickness."

Not sure if this is relevant to your quesiton, but the acyclic chromatic number is not bounded by geometric thickness $\overline{\theta}(G)$.