Approximation of holomorphic maps with a lower bound on the rank

Let $K$ be a closed polydisc or ball in $\mathbb{C}^n$, and let $Y$ be a quasi-projective algebraic manifold which is Zariski locally equivalent to $\mathbb{C}^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such a manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$, then every holomorphic map from a neighborhood of $K$ to $Y$ with rank $\ge r$ at every point of $K$ can be approximated uniformly on $K$ by entire maps $\mathbb{C}^n\to Y$ with rank $\ge r$ at every point of $\mathbb{C}^n$.