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In the present paper we prove that the $3$-fold projective tensor product of$c_0$, $c_{0} \widehat{\otimes}_\pi c_{0}\widehat{\otimes}_\pi c_{0}$, is notisomorphic to a subspace of $c_{0} \widehat{\otimes}_\pi c_{0}$. In particular,this settles the long-standing open problem of whether $c_{0}\widehat{\otimes}_\pi c_{0}$ is isomorphic to $c_{0} \widehat{\otimes}_\pic_{0}\widehat{\otimes}_\pi c_{0}$. The origin of this problem goes back to JoeDiestel who mentioned it in a private communication to the authors of paper"Unexpected subspaces of tensor products" published in 2006.

$c_{0} \widehat{\otimes}_\pi c_{0}\widehat{\otimes}_\pi c_{0}$ is not isomorphic to a subspace of $c_{0} \widehat{\otimes}_\pi c_{0}$