The norm conjecture for entropy of projections of symmetric log-concave vectors

Let $X$ be a symmetric log-concave random vector in $\mathbb{R}^n$. For $v\in\mathbb{R}^n$, define

$$N_X(v)= \begin{cases} e^{\mathcal{S}(\langle v,X\rangle)} & v\ne 0,\\ 0 & v=0. \end{cases}$$

The norm conjecture. The function $N_X$ defines a norm on $\mathbb{R}^n$. This would give a geometric formulation of entropy inequalities for one-dimensional projections of symmetric log-concave random vectors. The source does not provide evidence that the conjecture has been resolved.