Conjecture on the tempered positive-energy interpretation of the crossed-product K-group

Let $G$ be a semisimple Lie group with maximal compact subgroup $K$. Let $\tau_G^G$ and $\tau_K^K$ be the relevant twistings, let $\Lambda^*_{k-\textsf{h}^\vee}(K)$ be the indicated set of weights, and let $W_\mu$ and $V_\mu$ denote the corresponding positive energy representation and $K$-representation. Define

$$q:K_K^\bullet(K,\tau_K^K)\longrightarrow K_\bullet(\Gamma_0(\tau_G^G)\rtimes G).$$

Tempered positive-energy conjecture. The $K$-group $K_\bullet(\Gamma_0(\tau_G^G)\rtimes G)$ reflects the tempered, positive energy representations of $LG$ in a suitable sense, and the map $q$ is given by

$$q([W_\mu])=\operatorname{Ind}(\slashed{\partial}_{LG/K}\otimes V_\mu),\qquad \mu\in\Lambda^*_{k-\textsf{h}^\vee}(K).$$

The conjecture is motivated by the Freed–Hopkins–Teleman theorem, the stated theorem describing the reduced crossed-product $K$-group, and the associated commutative diagram. The source does not state a resolution.