Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of perfect complexes.

This question is about Hochschild cohomology in the dg category sense. The question in the title gives the gist, but a more precise question is the following. There is a restriction homomorphism $$ r\colon HH^\ast(D_{qcoh},D_{qcoh}) \to HH^\ast(D_{perf},D_{perf}). $$ This is clear if one computes Hochschild cohomology using the bar complex.

**Question:** Is $r$ an isomorphism?

The left-hand side makes me nervous because of the unbounded complexes, but I think this is true in the case of smooth $X$. I'm interested in singular varieties (or schemes). What about the affine case? To be concrete, what about the case where $X$ is a local complete intersection?

**Comments:** The category $D_{qcoh}$ looks much more ferocious than $D_{perf}$. The reason for bringing it into the picture is that its Hochschild cohomology seems to be better understood. One possible reference (which also discusses similar results for perfect complexes, but only in the smooth case) is
Toen's article on derived Morita equivalence. If I understood correctly (did I?), $HH^\ast(D_{qcoh},D_{qcoh})$ is isomorphic to
$$HH^\ast(X):= Ext^\ast_{\mathcal{O}_{X\times X}}(\delta_{\ast}\mathcal{O}_X,\delta_{\ast}\mathcal{O}_X)$$
where $\delta\colon X\to X\times X$ is the diagonal. And this is the thing I really want to compare to $HH^\ast(D_{perf},D_{perf})$, for the reason that I know how to compute it in examples. There is a local-to-global Ext spectral sequence converging to $HH^\ast(X)$, and for local complete intersections one can use the Hochschild-Kostant-Rosenberg isomorphism to understand the Ext-sheaf.

**Motivation:** I've been looking at manifestations of homological mirror symmetry in which one has an embedding of the Fukaya category of a symplectic manifold into $D_{per}$ for a mirror variety. I'd like to compute Hochschild cohomology of the Fukaya category via algebraic geometry.