Let $(X,\mathcal{O}_X)$ be a scheme (or more generally a ringed space). We know that in general the derived category of complexes of quasi-coherent modules $D(\text{Qcoh}(X))$ is not equivalent to the derived category of complexes of arbitrary $\mathcal{O}_X$-modules $D(X)$.
Nevertheless Thomason&Trobaugh's "Higher algebraic k-theory of schemes and of derived categories" B.16 and moreover SGA 6 tells us (if I understand it correctly) that
Let $X$ be either quasi-compact and semi-separated scheme, or else noetherian scheme. Then the natural functor $$ \phi: D_{+}(\text{Qcoh}(X))\rightarrow D_{+}(X) $$ is an equivalence, where $D_{+}(\bullet)$ is the derived category of complexes with bounded below cohomologies.
Now we turn to perfect complexes. For definition of perfect complex see this MO question How to prove that any perfect complex on an affine scheme is strictly perfect?
Let us denote $D_{\text{perf}}(\text{Qcoh}(X))$ the derived category of perfect complexes of quasi-coherent modules and $D_{\text{perf}}(X)$ the derived category of perfect complexes of any $\mathcal{O}_X$-modules.
By the above result in Thomason&Trobaugh, if $X$ is a quasi-compact and semi-separated scheme, or a noetherian scheme, then we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$.
$\textbf{My question}$ is: do we have a weaker condition on $X$ which still guarantees that $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?
