Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\mathcal{Y}$ are conglomerates and $f: \mathcal{X} \rightarrow \mathcal{Y}$ is a surjective map, then $f$ has a right inverse. 

The "Axiom of Choice for Classes" and the (usual) "Axiom of Choice for Sets"
may be defined in the obvious, analogous way (i.e., "Whenever $\mathcal{X}$ and $\mathcal{Y}$ are classes $\ldots$ etc." in the first case, and "Whenever $X$ and $Y$ are sets $\ldots$ etc." in the second).The Axiom of Choice for Classes implies the existence of a global choice function (meaning, a class function which is a choice function for the class of all non-empty sets), and the latter is, in turn, equivalent to the existence of a well-ordering of the universe. 

Well, it is well-known that the Axiom of Choice for Conglomerates implies that "Every category has a skeleton" (see, e.g., Adamék/Herrlich/Strecker book). On the other hand, Isbell and Wright have proved in the 60's that the statement "Every category has a skeleton" implies the existence of a well-ordering of the universe. 

My question is: considering the following assertions,

"The Axiom of Choice for Conglomerates"

and

"Every category has a skeleton"

are they equivalent statements ? At first glance, my conjecture is that the answer is "Yes", but I didn't find any reference for that. 

Added: In Freyd/Scedrov book it is shown that the Axiom of Choice for Sets is equivalent to the statement "Every small category has a skeleton". So I also wonder whether there is some ladder of equivalences between forms of the Axiom of Choice and assertions regarding the existence of skeletons for certain categories. I mean, considering statements as

"Every small category has a skeleton",

"Every locally small category has a skeleton", and

"Every category has a skeleton", and maybe other similar statements of this kind,

could we put each one of them in correspondence with some equivalent form of the Axiom of Choice ? The first one is equivalent to the Axiom of Choice for Sets, as I have just commented.