Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of $\mathcal{S}\mathcal{F}(\mathcal{A})$ consisting of DG modules homotopy equivalent to a direct summand of a finitely generated semi-free DG module over $\mathcal{A}$. Here, $\mathcal{S}\mathcal{F}(\mathcal{A})$ is the DG category of semi-free DG modules over $\mathcal{A}$, and a module is considered ``free'' if it is a direct sum of shifts of representable modules.

Example 1.9 of Lunts-Orlov is the assertion that the bounded derived category of $\mathcal{A}$ is (the homotopy category of) $Perf(\mathcal{A})$.

Under what conditions on $\mathcal{A}$ are all objects of $Perf(\mathcal{A})$ actually homotopy equivalent (isomorphic in $H_0$) to finitely generated semi-free DG modules?

The motivation of this question is the case when $\mathcal{A}$ is a strands algebra from bordered Heegaard Floer homology (the structure as a small dg category comes from the distinguished idempotents of $\mathcal{A}$).The papers that decategorify such $\mathcal{A}$ typically identify the unbounded derived category of $\mathcal{A}$ with $H_0$ of the category of Type D structures homotopy equivalent to operationally bounded Type D structures. The bounded derived category is identified with the full subcategory on Type D structures homotopy equivalent to finitely generated operationally bounded Type D structures.

An operationally bounded Type D structure is (more or less) the same as a semi-free dg module $X$ with a choice of decomposition of $X$ as $\mathcal{A} \otimes_{\mathcal{I}} N$ with respect to the $\mathcal{A}$-action, where $\mathcal{I}$ is the idempotent ring of $\mathcal{A}$ and $N$ is a left $\mathcal{I}$-module. Type D structures are isomorphic or homotopy equivalent iff the corresponding dg modules are isomorphic or homotopy equivalent. The same is true for finite generation.

So, based on the usual definition of perfect dg modules, I'd expect at first to have to use``Type D structures which split inject in $H_0$ into a finitely generated operationally bounded Type D structure'' as the bounded derived category. But this is less convenient for computing $K_0$, and I'd rather be able to replace an object of the bounded derived category with an isomorphic object which literally is a finitely generated bounded Type D structure.

In other words, my question is: are there any sufficient conditions one can place on dg algebras $\mathcal{A}$ for this to work, which are general enough to cover some interesting examples of $\mathcal{A}$ with nonzero differential?