The regular fine mixed subdivision 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 let a fine mixed subdivision be regular if it is induced by assigning heights to the vertices of the summands and projecting the lower facets of the resulting lifted Minkowski sum. For a basis $B$ of $\mathcal{T}_{n,d}$, say that the simplices of a subdivision are located at $B$ when their locations are exactly the elements of $B$. The regular fine mixed subdivision conjecture. For any basis $B$ of $\mathcal{T}_{n,d}$, there is a regular fine mixed subdivision of $n\Delta_{d-1}$ whose $n$ simplices are located at $B$. This is stated as a stronger converse to the preceding conjectural characterization, requiring every matroid basis to arise from a regular fine mixed subdivision; the source gives no resolution.