The Chow quotient and log canonical model conjecture for Grassmannians

Let $G(r,n)$ be the Grassmannian of $r$-planes in $\mathbb A^n$, let $H=\mathbb G_m^n$ be the standard diagonal torus, and let $G^0(r,n)$ be the open subset of $r$-planes projecting isomorphically onto every coordinate subspace $\mathbb A^I$ with $|I|=r$. Set

$$X(r,n)=G^0(r,n)/H\subset \overline{X}(r,n)=G(r,n)//H,$$

where $B$ denotes the boundary. The canonical duality identifies

$$\overline{X}(r,n)=\overline{X}(n-r,n).$$

Chow quotient and log canonical model conjecture. The inclusion $X(r,n)\subset \overline{X}(r,n)$ is the log canonical model precisely in the cases

$$(2,n),\ (3,6),\ (3,7),\ (3,8)$$

and those obtained from these by the canonical duality. Moreover, in these cases the pair $(\overline{X}(r,n),B)$ has toroidal singularities.

This conjecture refines the question of identifying the log canonical model of $X(r,n)$. The source notes that the Chow quotient and the alternative compactification do not generally agree, while the stated cases remain as positive-direction candidates.