Let $X$ be a scheme. Is the category $QCoh(X)$ of quasi-coherent sheaves on $X$ locally presentable? If so, can we say anything about the $\kappa$ for which $QCoh(X)$ is locally $\kappa$-presentable? (e.g. is it always finitely presentable? Or related to the $\kappa$ of Gabber's result?)
I'm particularly interested in the case where $X$ is quasi-comact quasi-separated (qcqs).
In my searching for references, I've come across answers ranging from "we don't know", "when qcqs", to "always", and would appreciate some clarity.