TITLE: KK-theory as a stable infinity-category and KU Mod
QUESTION [23 upvotes]: The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it carries a by now well-known triangulated category structure.
What seems like a step in the direction of establishing such a stable $\infty$-category structure is in the note 

Michael Joachim, Stephan Stolz, An enrichment of KK-theory over the category of symmetric spectra Münster J. of Math. 2 (2009), 143–182 (pdf)

which produces

an enrichment $\mathbb{KK}$ of $KK$ in symmetric spectra, in fact in KU-module spectra;
a symmetric monoidal enriched functor $\mathbb{KK} \to \mathrm{KU} Mod$.

(A partial equivariant generalization of this is given by Mitchener in arXiv:0711.2152.)
This prompts some evident questions:

Does this enrichment exhibit a presentation of a stable $\infty$-category structure (or close)?
How far is that functor from being homotopy full and faithful?

Has anyone thought about this? What can one say?
(I see that Mahanta has a note arXiv:1211.6576 along these lines, but not sure yet if it helps with KK.)

REPLY [2 votes]: Answering your first question in more generality: an $\infty$-category that is enriched over spectra is a stable $\infty$-category if it is closed under finite limits.