Let $(B,\pi)$ be an open book decomposition of a closed, connected, oriented 3-manifold $M$ with odd (even) binding number and with pages of Euler characteristic $\chi$. Is it possible to define another open book decomposition $(B',\pi')$ of $M$ with binding number one (two) and with pages of the same Euler characteristic $\chi$? Basically, I am trying to replace each pair of boundary components with a genus in pages of $(B,\pi)$ to obtain $(B',\pi')$.

I believe that this is too good to be true, but I have no candidate for a counter-example. However, I think one thing that might work is to pick the contact structure $\xi$ on $M$ compatible with $(B,\pi)$, and looking for $(B',\pi')$ in the collection of all open book decompositions compatible with $\xi$. Maybe if $\xi$ satisfies some certain properties (e.g. tightness), we can detect such a $(B',\pi')$. Yet I have no idea about how to argue this.

Is there any paper that (partially) answers this question or a similar one? Also, is there an obvious counter-example or proof for it (in the general case and in the case where $\xi$ is tight)?