The adelic amoeba half-space conjecture for lower-dimensional varieties

Let $X\subset\mathbb{G}_{\text{m}}^d$ be an irreducible variety of dimension $r$ such that its adelic amoeba $\mathscr{A}_{\mathbb{A}}(X)$ is not contained in any hyperplane. Let $L\subset\mathbb{R}^d$ be a linear subspace of codimension $r$, and let $H$ be a relatively open half-space in $L$ whose boundary contains $\mbf{0}$. Adelic amoeba half-space conjecture. One has

$$H\cap\mathscr{A}_{\mathbb{A}}(X)\ne\varnothing.$$

This conjecture extends the preceding theorem, which establishes the analogous assertion for hypersurfaces, to varieties of arbitrary lower dimension. Its status is not resolved in the supplied source.