Dolbeault S-duality matching conjecture for tempered magical triples

Let $\rho:\mathfrak{sl}_2\to\mathfrak{g}$ be a tempered magical $\mathfrak{sl}_2$-triple. Assume that $M_\rho$ is hyperspherical, equivalently $X_\rho$ has connected generic stabilizers, and that the representation $S_\rho\simeq V_\rho\oplus V_\rho^*$ of $\check{G}_\rho$ is of cotangent type. Let $G$ be the adjoint group with Lie algebra $\mathfrak{g}$, let $\check{G}$ be its Langlands dual group, and regard the Cayley morphism for $G^\mathbf{R}_\rho$ as a morphism of sheaves

$$A_{\mathrm{Dol}}(\phi):A_{\mathrm{Dol}}(M_\rho)\longrightarrow A_{\mathrm{Dol}}(M_\rho')\in\mathrm{QC}(\mathrm{Higgs}_G).$$

Dolbeault S-duality matching conjecture. Over the $\diamondsuit$-locus, the following matching data hold under $\mathbf{S}_{\mathrm{Dol}}$: the dual of $A_{\mathrm{Dol}}(M_\rho')$ is $B_{\mathrm{Dol}}(\check{M}_\rho')=\omega(\mathrm{Higgs}_{\check{G}_\rho})$, up to tensoring with a finite-rank vector bundle $W_\rho$ in the $\ast$-cases; the dual of $A_{\mathrm{Dol}}(M_\rho)$ is

$$B_{\mathrm{Dol}}(\check{M}_\rho)=\bigoplus_{k\geq 0}(\operatorname{Sym}^k\mathrm{DH}_{V_\rho})[k],$$

again tensored with $W_\rho$ in the $\ast$-cases; and the dual of $A_{\mathrm{Dol}}(\phi)$ is the natural morphism $B_{\mathrm{Dol}}(\check{M}_\rho)\to B_{\mathrm{Dol}}(\check{M}_\rho')$ induced by projection of the Dirac--Higgs bundle to its zero section,

$$\mathrm{DH}_{V_\rho}\longrightarrow\mathrm{Higgs}_{\check{G}_\rho},$$

with the same tensoring convention. The claim is a proposed matching under relative Langlands/S-duality over the specified locus; no resolution is supplied in the source.