Cotangent-case Borel-orbit duality conjecture

Let $G$ be a reductive group with Langlands dual group $G^\vee$, and let $W$ be the Weyl group. Let $X$ be a smooth spherical $G$-variety such that $T^*X$ is hyperspherical and has relative Langlands dual of the form $T^*X^\vee$ for a smooth spherical $G^\vee$-variety $X^\vee$. Let $B\backslash X$ and $B^\vee\backslash X^\vee$ denote the sets of Borel orbits, and let $\operatorname{rank}(Y)$ denote the dimension of the lattice of $B$-semi-invariant rational-function characters on a variety $Y$. Cotangent-case orbit-duality conjecture. There exists a bijection

$$\Phi:B\backslash X\longrightarrow B^\vee\backslash X^\vee$$

that intertwines Knop's Weyl-group action and satisfies $\operatorname{rank}(x)+\operatorname{rank}(\Phi(x))=\operatorname{rank}(G)$. For every simple reflection $s\in W$, an orbit $x\in B\backslash X$ has type $G$ with respect to $s$ if and only if $\Phi(x)$ has type $T2$ with respect to $s$, and conversely.

This conjecture translates the relative Langlands duality prediction into combinatorics of Borel orbits. The source proposes it based on computations used in proving the main theorem; no resolution is given.