Has anyone worked out the universal property of the Cauchy completion of an enriched category?

What I have in mind is a definition that makes sense in any (suitable) framed bicategory, such as $\mathcal{V}$-Prof. Is there a notion of when an object in a framed bicategory has all absolute limits, formulated in terms of something like absolute Kan extensions? If so, is there a definition of Cauchy completion in these terms?

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