Yun's uniqueness conjecture for minimal reduction types

Let $G$ be the reductive group underlying the affine Grassmannian, let $L^{\heartsuit}{\mathfrak g}$ be the space of elements under consideration, and for $\gamma\in L^{\heartsuit}{\mathfrak g}$ let $\operatorname{RT}_{\min}(\gamma)$ be the set of minimal nilpotent orbits, under the closure order, whose affine Springer fibers over those orbits are non-empty. Yun's conjecture. For any $\gamma\in L^{\heartsuit}{\mathfrak g}$, the minimal reduction set $\operatorname{RT}_{\min}(\gamma)$ consists of a single nilpotent orbit. The conjecture asserts uniqueness of the minimal reduction type beyond the shallow elements for which Yun proved uniqueness and constructed the minimal reduction map; the supplied source does not state whether the conjecture has been resolved.