Invariant subspaces for Fr\'echet spaces without continuous norm

Let $(X,(p_j))$ be a Fr\'echet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and only if there exists $j_0\ge 1$ such that $\ker p_{j+1}$ is of finite codimension in $\ker p_{j}$ for every $j\ge j_0$.