Higher projective tensor products of $c_0$

Let $m,n$ be positive integers with $m<n$. Under certain assumptions on the Banach space $X$, we prove that the $n$-fold projective tensor product of $X$, $\widehat{\otimes}^n_\pi X$, is not isomorphic to any subspace of any quotient of the $m$-fold projective tensor product, $\widehat{\otimes}_\pi^m X$. In particular, we prove that $\widehat{\otimes}^n_\pi c_0$ is not isomorphic to any subspace of any quotient of $\widehat{\otimes}_\pi^m c_0$.