I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe that all sheaves in the unbounded category of cochains $\operatorname{Ch}(X)=\operatorname{Ch}(\operatorname{QCoh}(X))$ admit a $K$-flat resolution, in the sense of Spaltenstein.
Is it the case, however, that $\operatorname{Ch}(X)$ admits functorial$K$-flat resolutions?
Though I'm open to any information in this regard, here I might specifically seek an endofunctor\begin{equation}T:\operatorname{Ch}(X)\to \operatorname{Ch}(X)\end{equation}which takes $K$-flat values and comes equipped with a natural transformation $\epsilon_M:T(M)\to M$ which is a quasi-isomorphism at all $M$. Or, maybe more reasonably, I might seek such an endofunctor $T:\mathscr{K}(X)\to \mathscr{K}(X)$ and transformation at the level of the homotopy $\infty$-category.
In the presence of a suitable model structure I believe that such things would exist, but I cant find any specific information in this regard in the literature.
