The fine mixed subdivision basis conjecture for the matroid $\mathcal{T}_{n,d}$

Let $\mathcal{T}_{n,d}$ be the matroid introduced from the combinatorial encoding of arrangements of $d$ generic complete flags in $\mathbb{C}^n$. Let $n\Delta_{d-1}$ be the Minkowski sum of $n$ copies of the simplex $\Delta_{d-1}$, and consider the fine mixed subdivisions of this polytope. The locations of the simplices in such a subdivision form subsets of the ground set of $\mathcal{T}_{n,d}$. The fine mixed subdivision basis conjecture. The possible locations of the simplices in a fine mixed subdivision of $n\Delta_{d-1}$ are precisely the bases of the matroid $\mathcal{T}_{n,d}$. This would extend the established connection between fine mixed subdivisions, triangulations of products of simplices, and the matroid of generic flag arrangements; the source presents it as a conjectural generalization, with no resolution stated.