Should the PM establish friendship bonds with project stakeholders?

As a project manager you also have to be able to say "no" to stakeholders - in fact, the best answer is "yes, but it will cost this much, will take this much time etc." A personal friendship may make this difficult, hence the potential conflict of interests indicated in previous comments.

Moreover, even if the friendship does not really impact the professional relationship, the other stakeholders may be under the impression that it does - hence "perceived conflict of interest".

So my guideline is "great professional relationship - OK, but friendship - after the project ends".

I agree. That's the way I see it, too.
And I don't like friendship with every one - even if I have a very good professional relationship with him or her.
p.s.: I think the term friendship has different meanings: to go to a bar, do some activities together - that's o.k. even during project, as Maria wrote - I call that "low friendship", sometimes not even "friendship": in German we have a separate term for that: "guter Bekannter". To share privacy, like invitation to home etc. for me is another level - and what I call "real friendship" - and for that I use the German word "Freundschaft".