Relative Humphreys conjecture for antispherical cells

Let $G$, $G_1$, $W$, ${}^{\mathrm{f}}W^{\mathrm{f}}$, $\mathcal{C}_p$, $\iota_{\Bbbk}$, and the corresponding nilpotent orbit $\mathscr{O}^{\mathbb{C}}_{\mathbf{c}(w)}$ be as in the paper. For $w\in{}^{\mathrm{f}}W^{\mathrm{f}}$, let $\overline{V}_{G_1}(\mathsf{T}(w\cdot_p0))$ denote the relevant reduced support variety. Relative Humphreys conjecture. For every $w\in{}^{\mathrm{f}}W^{\mathrm{f}}$,

$$\overline{V}_{G_1}(\mathsf{T}(w\cdot_p0))=\overline{\iota_{\Bbbk}(\mathscr{O}^{\mathbb{C}}_{\mathbf{c}(w)})}.$$

This is presented as a relative version of the regular Humphreys conjecture, relating support varieties of tilting modules indexed by doubly minimal affine Weyl-group elements to nilpotent-orbit closures. The source does not provide a resolution status for this relative statement.