There are two notions of completions of slightly different nature, and I am wondering if there is a precise statement relating them.

The first one is the “homotopical” (or maybe it should be called algebraic) completion of a morphism in a sense of, e.g. the paper of Kathryn Hess (section 4), i.e. totalization of a monadic bar-construction.

The second one is the “geometric completion” of a morphism of schemes in the sense of Gaitsgory—Rozenblyum, see for instance this paper section 6. Here the completion is calculated using the fiber product $X^{\wedge}_Y= X\times_{X_{\textrm{dR}}} Y_{\textrm{dR}}.$

Is there a way to relate the two in the case when the morphism of derived schemes is representable? More precisely, I am looking for a statement of the type: functions on the geometric completion is the homotopical completion of the morphism of underlying rings. I would also really appreciate a reference if there exists one.