The planarity and semi-dilation bounds for open books

Let $Y$ be an open book with page $W$. Here $\operatorname{P}$ denotes planarity and $\operatorname{SD}$ denotes semi-dilation order.

The open-book conjecture.

$$\operatorname{P}(Y)\le \operatorname{P}(W),\qquad \operatorname{SD}(Y)\le \operatorname{SD}(W).$$

This would extend the upper-bound theorem proved in the paper from the particular cases treated there to arbitrary open books, relating contact invariants of the open book to those of its page.