Rigidification conjecture for the dual of the adjoint bundle spectrum

Let $G$ be the compact Lie group and $EG$ the universal principal $G$-bundle, with $Ad(EG)$ its adjoint bundle and $T_v$ its vertical tangent bundle. Let $LBG^{-TBG}$ denote the associated pro-ring spectrum, and let the Spanier–Whitehead dual be taken in the model category of pro-spectra described by Christensen and Isaksen. Rigidification conjecture. There is a rigidification of $LBG^{-TBG}$ which is weakly equivalent in that model category to the Spanier–Whitehead dual of $Ad(EG)^{-T_v}$. This would make the weak Spanier–Whitehead duality suggested by the preceding theorem into a rigidified statement in the model category of pro-spectra; the source does not provide evidence that the conjecture has been resolved.