The fp-injective module conjecture

Let $A$ be a ring, and let $\operatorname{Fpinj} A$ be the category of fp-injective $A$-modules. Write $\mathbf K(\operatorname{Inj} A)$ for the homotopy category of complexes of injective $A$-modules, $\mathbf K(\operatorname{Fpinj} A)$ for the homotopy category of complexes in $\operatorname{Fpinj} A$, and $\mathbf D(\operatorname{Fpinj} A)$ for its derived category. If $A$ is coherent, also write $\mathbf D_{\mathrm{pur}}(\operatorname{Mod} A)$ for the pure derived category, $\mathbf D(\operatorname{Fpinj} A)^c$ for the compact objects, and $\mathbf D^b(\operatorname{mod} A)$ for the bounded derived category of finitely presented $A$-modules. The fp-injective module conjecture. For every ring $A$, the composite

$$\mathbf K(\operatorname{Inj} A)\xrightarrow{\mathrm{inc}}\mathbf K(\operatorname{Fpinj} A)\xrightarrow{\mathrm{can}}\mathbf D(\operatorname{Fpinj} A)$$

is an equivalence. If $A$ is coherent, then $\mathbf D(\operatorname{Fpinj} A)$ is compactly generated, and the composite

$$\mathbf D(\operatorname{Fpinj} A)\to\mathbf D_{\mathrm{pur}}(\operatorname{Mod} A)\to\mathbf D(\operatorname{Mod} A)$$

induces an equivalence

$$\mathbf D(\operatorname{Fpinj} A)^c\xrightarrow{\sim}\mathbf D^b(\operatorname{mod} A).$$

This conjecture extends analogous results for flat modules and predicts that injective resolutions model the derived category of fp-injective modules. Its general categorical form is stated in terms of exact functor categories, but the source provides no resolution of either formulation.