The simplex conjecture for random simplex volumes

Let $K$ be a convex body in $\mathbb{R}^d$, let $T^d$ be a $d$-dimensional simplex, let $p\geqslant 1$, and let $n\geqslant d+1$. Write $\mathbb{E}^p_*(K)$ and $\mathbb{E}^p_n(K)$ for the random-simplex volume functionals defined in the paper. Simplex conjecture. One has

$$\mathbb{E}^p_*(K)\leqslant\mathbb{E}^p_*(T^d)\qquad\text{and}\qquad \mathbb{E}^p_n(K)\leqslant\mathbb{E}^p_n(T^d),$$

with equality if and only if $K$ is a simplex. This is presented as the most general conjecture governing the maximum of the random-simplex functionals; the source does not state whether it has been resolved.