For sets $A,B$, let $|A|\leq^*|B|$ say that there exists an onto map $f:B\rightarrow A$ or $A=\emptyset$.
My question is, is $$\forall A,B(|A|\leq^*|B|\longrightarrow |A|\leq|B|)$$ equivalent to the axiom of choice?
We don't know.
This is known as the partition principle (note that the other implication is trivial in $\sf ZF$). The problem whether or not this is equivalent to the axiom of choice has been open for over a century now.
There isn't much to say about it, really. We know it implies some basic choice principles such as "Every infinite set is Dedekind infinite", but we don't know a lot more. Every time I think about the problem I run into the same problem, we don't have enough tools to manage - or even understand - the structures of cardinals in arbitrary models of $\sf ZF$. Not even under $\leq$ and let alone under $\leq^*$.