TITLE: Are triangulated equivalence detected at compact level?
QUESTION [15 upvotes]: Suppose that $D$ and $E$ are compactly generated triangulated categories, even algebraic (i.e. equivalent to derived categories of small dg categories) if we want, and asume that their subcategories $D^c$ and $E^c$ of compact objects are triangle equivalent. Are $D$ and $E$ triangle equivalent?. By Theorem 9.2 of Keller's 'Deriving dg categories', we know that the answer is 'yes' when either $D$ or $E$ is the derived category of (the category of modules over) a small $k$-linear category, but I do not know the general answer. 
Any help would be highly appreciated.

REPLY [6 votes]: Consider the case where $D$ is the ordinary stable homotopy category of spectra, and $E$ is assumed to have all coproducts (which is clearly a necessary condition), and we also assume that $D^c\simeq E^c$ as tensor triangulated categories.  It is an old conjecture of Margolis that this forces $D$ to be equivalent to $E$, and I believe that this is still open in full generality.  In both $D$ and $E$ one can define a square-zero ideal of phantom maps.  Hovey, Palmieri and I proved that $D/\text{Ph}_D\simeq E/\text{Ph}_E$ and that $\text{Ph}_D\simeq\text{Ph}_E$ as a module over this quotient.  Schwede and Shipley proved in https://arxiv.org/abs/math/0108143 that $D\simeq E$ provided that $E\simeq\text{Ho}(E_0)$ for some Quillen model category $E_0$.