Busemann-type norm conjecture for symmetric $\kappa$-concave measures

Fix $\kappa\in[-\infty,\frac1d]$. Let $(U,V)$ be the coordinates of a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$. Symmetric $\kappa$-concave Busemann conjecture. For every $p\in[1,\infty]$, whenever all quantities are finite,

$$N_p^{1/2}(U+V)\leq N_p^{1/2}(U)+N_p^{1/2}(V).$$

Equivalently, if $X$ is a symmetric $\kappa$-concave random vector in $\mathbb{R}^d$, then for any $p\in[1,\infty]$ the function

$$M_p^X(v)=\begin{cases}N_p^{1/2}(\langle v,X\rangle)&v\neq0,\\0&v=0\end{cases}$$

defines a norm on $\mathbb{R}^d$ when it is finite everywhere. This conjecture is intended to interpolate between the known $p=\infty$ result and the Shannon case, while subsuming the other results and conjectures in the section. Its general status is open.