It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the fact that Reihardt cardinals are inconsistent with choice already makes them, in a way, "wrong", that is, I'd believe personally that Reinhardt cardinals cannot exist in $V$ anyways. However, I have never seen any claims like this. So my question is:

Is it believed that Reinhardt cardinals can be consistent without choice? Are there any solid arguments for/against them? (other than "we haven't found them inconsistent, so they should be consistent")

Thanks in advance.

Note: I added soft-question tag because this question is more about philosophical views. Feel free to remove it if you think otherwise.