Ball–Nayar–Tkocz Busemann norm conjecture for Shannon entropy

Let $X$ be a symmetric log-concave random vector in $\mathbb{R}^d$. Define

$$M_1^X(v)=\begin{cases}N_1^{1/2}(\langle v,X\rangle)&v\neq0,\\0&v=0.\end{cases}$$

Ball–Nayar–Tkocz's conjecture. The function $M_1^X$ defines a norm on $\mathbb{R}^d$. The conjecture is the Shannon-entropy case of a Busemann-type theorem. The source records a later result proving the corresponding entropy inequality with exponent $\alpha=1/5$, so the conjecture is treated as resolved.