Separate subharmonicity conjecture

Let $F:\mathbb{R}^{2d}\to\mathbb{R}$ be a Lipschitz, positively homogeneous function of order one. For each $j=1,\ldots,d$, consider the pair of variables $(x_j,x_{j+d})$.

Separate subharmonicity conjecture. If $F$ is subharmonic with respect to every pair $(x_j,x_{j+d})$, then $F$ is non-negative. This geometric statement is intended to provide the key ingredient in the proof of the anisotropic Ornstein non-inequality conjecture.