Saroglou's lower-bound conjecture for the second projection body

Let $K$ be a $d$-dimensional convex body containing the origin in its interior. Define

$$m(K):=\min_{x\in S^{d-1}}\frac{h_{\Pi^2K}(x)}{h_K(x)V(K)^{d-2}}.$$

For $x\in S^{d-1}$, let $K\cap(sx+x^\bot)$ be the $(d-1)$-dimensional section and let $V_{d-1}$ denote its volume. Saroglou's lower-bound conjecture.

$$m(K)\geq\max_{x\in S^{d-1}}\omega_{d-2}^{d-1}\omega_{d-1}^{3-d}\left\{\frac{\left(\displaystyle\int_{-h_K(x)}^{h_K(x)}V_{d-1}\left(K\cap(sx+x^\bot)\right)^{\frac{d-2}{d-1}}ds\right)^{d-1}}{h_K(x)V(K)^{d-2}}\right\}.$$

The source presents this as a conjectural all-dimensional strengthening related to its proved three-dimensional lower-bound results; no resolution is given.