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.