On the conjectured capillary Blaschke-Santaló inequality

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