Taut foliations and an open book for the negative contact structure

Suppose that $(S,\varphi)$ is an open book with connected binding for a contact 3-manifold $(Y,\xi)$, and suppose the fractional Dehn twist coefficient of $\varphi$ is bigger than 1.

Roberts proved in this case that $Y$ admits a co-orientable taut foliation which is transverse to the binding. Eliashberg--Thurston's construction (together with its improvements by Bowden and Roberts--Kazez) therefore produces tight contact structures $\xi_\pm$ on $\pm Y$ which are weakly semi-filled by a symplectic form on $Y\times [0,1]$. Honda--Kazez--Matic proved moreover that $\xi = \xi_+$ for one of these Roberts-type foliations. 

Problem: Find a way of producing from $(S,\varphi)$ an open book supporting the other contact structure $(-Y,\xi_-)$.  

One has to be a bit careful about how to interpret this because it's not a priori true that $\xi_-$ is even an invariant of the foliation (two negative perturbations of the foliation may not be isotopic).

Note that $(S,\varphi^{-1})$ is an open book for $-Y$, but cannot support $\xi_-$ since $\xi_-$ is tight while $\varphi^{-1}$ is not right-veering. 

This question is perhaps particularly interesting in light of Massoni's recent results about reconstructing taut foliations from positive and negative contact structures on a 3-manifold.

Update: Rithwick Vidyarthi has a much simpler way of proving Honda--Kazez--Matic's theorem, based on work of Etnyre--Van Horn Morris, which could be helpful for the problem above.

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