Connes–Karoubi regulator conjecture for generalized Dirac operators

Let $X$ be a closed Riemannian manifold of odd dimension $d$ equipped with a generalized Dirac operator $\slashed{D}$. Let $\sigma_{d+1}:K_{d+1}(C^{\infty}(X))\to {\mathbf{ku}}\mathbb{C}/\mathbb{Z}^{-d-2}(X)$ be the regulator map, let $b_{\slashed{D}}:C^{\infty}(X)\to \mathcal{M}^{d}$ classify the Fredholm module of $\slashed{D}$, and let $\delta:K_{d+1}(\mathcal{M}^{d})\to \mathbb{C}/\mathbb{Z}$ be the Connes–Karoubi multiplicative character. Define $r_{\slashed{D}}$ by pairing with the $K$-homology class of $\slashed{D}$. Connes–Karoubi regulator conjecture. The diagram with maps $\sigma_{d+1}$, $b_{\slashed{D}}$, $r_{\slashed{D}}$, and $\delta$ commutes; equivalently, $r_{\slashed{D}}\circ\sigma_{d+1}=\delta\circ K_{d+1}(b_{\slashed{D}})$. The conjecture is supported by a result proving the equality on the subgroup of topologically trivial classes. Its general case is not resolved in the supplied text.