Saroglou's three-dimensional ellipsoid conjecture for a section functional

Let $K$ be a symmetric convex body in $\mathbb{R}^3$, let $h_K$ be its support function, let $V(K)$ be its volume, and let $V_2$ denote planar area. For $x\in S^2$, write $K\cap(sx+x^\bot)$ for the planar section at signed distance $s$. Saroglou's section-functional conjecture. The quantity

$$\max_{x\in S^2}\left\{\frac{\left(\displaystyle\int_{-h_K(x)}^{h_K(x)}V_2\left(K\cap(sx+x^\bot)\right)^{\frac12}ds\right)^2}{h_K(x)V(K)}\right\}$$

is minimized precisely for three-dimensional ellipsoids. The source gives a related lower bound theorem but leaves this proposed sharp minimization statement as a conjecture.