On the conjectured capillary Blaschke-Santal\'o inequality

We prove that the conjectured capillary Blaschke-Santal\'o inequality holds for any unconditional, strictly convex capillary hypersurface when $\theta \in \left(0, \tfrac{\pi}{2}\right)$. Moreover, for $\theta \in \left(\tfrac{\pi}{2}, \pi\right)$, we show that the capillary volume product has no finite upper bound.