Conjectural Mori chamber decomposition of the one-point blow-up of a Grassmannian

Let $\mathbb G(r,n)_1$ be the blow-up of the Grassmannian $\mathbb G(r,n)$ at one point, let $H$ denote the pullback of the hyperplane class, and let $E$ denote the exceptional divisor. Write $\operatorname{Mov}(\mathbb G(r,n)_1)$ for the movable cone and $\operatorname{Eff}(\mathbb G(r,n)_1)$ for the effective cone. Conjecture on the Mori chamber decomposition. The movable cone is

$$\operatorname{Mov}(\mathbb G(r,n)_1)=\begin{cases} \operatorname{cone}(H,H-rE) &\text{if } n=2r+1,\\ \operatorname{cone}(H,H-(r+1)E) &\text{if } n>2r+1,\end{cases}$$

and $E,H,H-E,\dots,H-(r+1)E$ are the walls of the Mori chamber decomposition of $\operatorname{Eff}(\mathbb G(r,n)_1)$. This is presented as a conjectural description addressing the Mori chamber decomposition problem; no resolution evidence is supplied in the excerpt.