Log-Minkowski inequality for centrally symmetric convex bodies

Let $C,D\in\mathrm{Conv}_s(\mathbb{R}^{n+1})$ be centrally symmetric convex bodies of unit volume, and let $h_C,h_D$ denote their support functions. Let $\nu_C$ be the cone measure of $C$. Log-Minkowski inequality. One has

$$\int \log\left(\frac{h_D}{h_C}\right)\,d\nu_C\geq 0.$$

The inequality is related to an equivalent log-Brunn–Minkowski inequality. The source attributes it to prior work but gives no evidence of a resolution.