This precise question was phrased as the vanishing conjecture in Hochschild homology and semiorthogonal decompositions. But we now know that there exist so called (quasi)phantom categories, which give counterexamples. These are categories with vanishing Hochschild homology, and vanishing (resp. torsion) Grothendieck group. As they are admissible subcategories of derived categories of smooth projective varieties, they have all nice properties you could want for their dg enhancements. An overview of some constructions:
Gorchinskiy, Sergey; Orlov, Dmitri, Geometric phantom categories, Publ. Math., Inst. Hautes Étud. Sci. 117, 329-349 (2013). ZBL1285.14018.
Böhning, Christian; Graf von Bothmer, Hans-Christian; Katzarkov, Ludmil; Sosna, Pawel, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS) 17, No. 7, 1569-1592 (2015). ZBL1323.14014.
Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum’s fake projective planes, Adv. Math. 278, 238-253 (2015). ZBL1327.14081.
Galkin, Sergey; Shinder, Evgeny, Exceptional collections of line bundles on the Beauville surface, Adv. Math. 244, 1033-1050 (2013). ZBL1408.14068.
Böhning, Christian; Graf von Bothmer, Hans-Christian; Sosna, Pawel, On the derived category of the classical Godeaux surface, Adv. Math. 243, 203-231 (2013). ZBL1299.14015.
Alexeev, Valery; Orlov, Dmitri, Derived categories of Burniat surfaces and exceptional collections, Math. Ann. 357, No. 2, 743-759 (2013). ZBL1282.14030.
What is interesting is that Hochschild cohomology can detect their non-vanishing, and all kinds of interesting behavior regarding deformation theory arises, see
Kuznetsov, Alexander, Height of exceptional collections and Hochschild cohomology of quasiphantom categories, J. Reine Angew. Math. 708, 213-243 (2015). ZBL1331.14024.