This paper deals with a matrix pencil completion problem. Specifically, the author gives explicit, necessary and sufficient conditions for a completion of an arbitrary matrix pencil up to a pencil with a prescribed set of Kronecker invariants, in the case when the only nontrivial invariants of the resulting pencil are its column (or equivalently row) minimal indices. In a previous paper, the author described the possible Kronecker invariants of a matrix pencil with a prescribed subpencil which has only column (row) minimal indices as nontrivial Kronecker invariants. In this paper, she solves the dual problem over an arbitrary field \(\mathbb{F}\), giving complete and explicit solution of the following problem: Let \(A(\lambda) \in \mathbb{F}[\lambda]^{(n+p) \times (n+m)}\) and \(M(\lambda) \in \mathbb{F}[\lambda]^{(n+p+l) \times (n+m+k)}\) be matrix pencils, such that the set of nontrivial Kronecker invariants of \(M(\lambda)\) consists only of its column (row) minimal indices. Find necessary and sufficient conditions for the existence of pencils \(X(\lambda) \in \mathbb{F}[\lambda]^{(n+p) \times k}\), \(Y(\lambda) \in \mathbb{F}[\lambda]^{l \times (n+m)}\) and \(Z(\lambda) \in \mathbb{F}[\lambda]^{l \times k}\) such that the pencil \[ \left[ \begin{matrix} A(\lambda) & X(\lambda) \\ Y(\lambda) & Z(\lambda) \end{matrix} \right], \] is strictly equivalent to \(M(\lambda)\).