Existence of a singular open book without radiality

Let $G$ satisfy the conditions of Theorem sf2, except that its discriminant need not be radial. For sufficiently small $\varepsilon>0$, consider the sphere $S^{m-1}_{\varepsilon}$ and the discriminant preimage $G^{-1}(\operatorname{Disc} G)$. Open-book conjecture. There is a singular open book structure on

$$S^{m-1}_{\varepsilon}\setminus G^{-1}(\operatorname{Disc} G),$$

independent of the sufficiently small choice of $\varepsilon$. This would extend the sphere-fibration existence result beyond the radiality assumption; the source gives no resolution of the conjecture.