A sharp Santaló inequality in a slab with centrally symmetric sections

We prove a sharp Santaló-type inequality for the product of $V(φ)$ and $V$ of the Legendre transform of $φ$, where $φ$ is an even geometric convex function on $\mathbb{R}^n$ and $V(φ)=\int(1+φ)^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the same volume,