Braid-arrangement minimizer conjecture for matroid fans

Let $M$ be a loopless matroid on a finite ground set $E$, and let $4\Sigma(M) \subseteq \mathbb{R}^E$ be its matroid fan. For a rational subspace $R \subseteq \mathbb{R}^E$, define

$$\operatorname{adim}(M):=\min\{2\dim_{\mathbb{R}}(\Sigma(M)+R)-\dim_{\mathbb{R}}(R)\mid R\subseteq\mathbb{R}^E\text{ is rational}\}.$$

Here a rational subspace is spanned by its intersection with $\mathbb{Q}^E$, and $\Sigma(M)+R$ denotes the Minkowski sum. Braid-arrangement minimizer conjecture. The minimum is attained by some subspace $R$ in the braid arrangement, namely an intersection of hyperplanes of the form $x_i=x_j$ for distinct $i,j\in E$.

This conjecture would make the matroidal formula for amoeba dimension more explicit by restricting the minimization to braid-arrangement subspaces. The paper presents its theorem as being inspired by this conjecture; no resolution is supplied in the given text.