A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the following . Let $X$ be a Hausdorff topological space. Then, $X$ is compact if and only if $X^{\kappa}$ is Lindelöf for any cardinal $\kappa$.

If the above is indeed a fact, can one restrict the class of $\kappa$'s for which the characterization is still valid?

Note: Here I'm thinking under ZFC.