Ball–Nayar–Tkocz planar Shannon reverse entropy power conjecture

Let $(U,V)$ be the coordinates of a symmetric log-concave random vector in $\mathbb{R}^2$. Ball–Nayar–Tkocz's conjecture.

$$N_1^{1/2}(U+V)\leq N_1^{1/2}(U)+N_1^{1/2}(V).$$

This inequality is equivalent to the triangle inequality needed for the associated Shannon-entropy Busemann function to define a norm. The source marks the conjecture as resolved through the cited work of Ball, Nayar and Tkocz.