The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.

However all the papers that I have seen dealing with this problem or variants of this problem say things like "It is a longstanding conjecture that a transitive projective plane is Desarguesian" or "It has been conjectured that ..." and none of them actually say who made the original conjecture.

I've looked in Kantor's papers on flag-transitive planes, Dembowski's book on Finite Geometries, Ostrom and Wagner's paper proving that planes with *doubly* transitive groups are Desarguesian and Higman and McLaughlin's paper on ABA groups.

So is the conjecture folklore? Or can anybody point me to an explicit reference?

EDIT: Question has been up for a few days without answer so I'm giving up and assigning the result to "folklore". In the meantime, I've written a blog post about it for posterity: http://symomega.wordpress.com/2011/06/03/an-elusive-conjecture/