This question is closely related to Grothendieck's "probleme de topologies" which was solved by Jari Taskinen in 1986. As far as I have heard, Susanne Dierolf (the advisor of my thesis) then observed that his construction also yields a negative solution of the problem you mention. I do not know if Dierolf published this somewhere. The following reference should be helpful: Jari Taskinen, The projective tensor product of Frechet-Montel spaces. Studia Math. 91 (1988), 17–30.
Jochen Wengenroth
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