The following question came up in a discussion the other day and I have been wondering whether something is known about it. Everything below takes place over $\mathbb{C}$. I don't have the expertise to know if this is trivial or of interest. Suppose a commutative dga has a free-commutative model $(\wedge V , d)$ where V is a finite dimensional vector space.

Recall that $T^{poly}$ is the Lie-algebra of polyvector fields on $\wedge V$ (yes, everything is superized as V will be in general graded) with Schouten bracket. Part of Kontsevich's formality theorem says that the HKR map $ T^{poly} \to HC^*$(Hochschild cochains) is the first Taylor coefficient in an $L_\infty$ quasi-isomorphism between the two.

We can think of the derivation $d$ as corresponding to a vector-field $v$. It follows from a spectral sequence argument that the HKR map gives a quasi-isomorphism: $$ (T^{poly},[v,-]) \to HC( \wedge V,d)$$

Question: Can this map be upgraded to a map of $L_\infty$ algebras?

Certainly, the Taylor coefficients in the usual formality map must be doctored.

A related statement that does seem to be true and standard is that there is an $L_\infty$ quasi-isomorphism $(T^{poly}[[t]],[tv,-]) \to HC^*(\wedge V[[t]],td )$ Thus, the question is in some reasonable sense about convergence of this isomorphism. Maybe one can prove the claim by a close inspection of Kontsevich's integral formulas. Based upon these facts, however, it seems plausible to me that that the statement is in general false, but I was unable to come up with a counterexamples or an a priori reason (I didn't try too hard however). Is it true for some more restrictive group of commutative dg algebras, for example pure Sullivan algebras?

Update: Having finally looked at the Kontsevich formulas, I'm beginning to think there are some simple counting reasons that make the above formula converge, but am not sure that $f_1$ stays the same (though I believe it remains a quasi-iso). Any confirmation or help would be great. Otherwise, I'll keep thinking and update again.