Coarsest Dressian structure conjecture for tropical Grassmannians

Let $\operatorname{TGr}_0(k,n)$ denote the characteristic-zero tropical Grassmannian, let $\operatorname{Dr}(k,n)$ denote the Dressian, and let $\mathcal I_{n,k}$ be the Plücker ideal. For $w,v\in\operatorname{TGr}_0(k,n)$ in the relative interior of a maximal Gröbner cone, write $(\cdot):p^\infty$ for saturation at the product $p$ of all Plücker variables. Coarsest Dressian structure conjecture. For any such $w$ and $v$,

$$w \text{ and } v \text{ lie on the same cone of } \operatorname{Dr}(k,n) \quad \Longleftrightarrow\quad \operatorname{in}_w(\mathcal I_{n,k}):p^\infty = \operatorname{in}_v(\mathcal I_{n,k}):p^\infty.$$

In particular, there is a subfan of the Dressian $\operatorname{Dr}(n,k)$ which coarsens the Gröbner subfan supported on $\operatorname{TGr}_0(n,k)$. This conjecture concerns the existence of a natural coarsest polyhedral structure on tropical Grassmannians; all computations up to and including the paper's support it, but its general validity remains open.