Criterion for quasi-affineness of abstract horospherical stacks

Let $G$ be a connected reductive algebraic group. Let $\mathcal{X}$ be a smooth integral finite type algebraic stack over $k$ with affine diagonal, $\operatorname{Pic}(\mathcal{X})=0$, and diagonalizable inertia groups. Suppose that $\mathcal{X}$ contains a big open substack $\mathcal{Y}$, meaning that its complement has codimension at least $2$. If $\mathcal{Y}$ is a smooth quasi-affine scheme and $\mathcal{X}$ is an abstract horospherical $G$-stack, then $\mathcal{X}$ is a quasi-affine scheme.

Criterion for quasi-affineness. Under these hypotheses, $\mathcal{X}$ is a quasi-affine scheme.

This conjectural criterion is used to obtain a structure result for smooth, not necessarily toroidal, abstract horospherical stacks. Its status is not resolved in the supplied source.