Degree inequality of a polynomial map distinguishing hyperplanes

Let $H_1, ldots, H_n$ be $n$ linearly independent hyperplanes in $k^n$, for some arbitrary field $k$. Let $X = H_1 cup H_2 cup cdots cup H_n$. Is it true that if $F=(f_1, ldots, f_n)$ is a polynomial map from $k^n$ to $k^n$, such that $F(X) cap F(k^n – X) = emptyset$, then $sum deg(f_i) ge n$?

This holds under the stronger condition that for all $a in X$, $F(a)$ has at least one coordinate equal to zero, and for all $a notin X$, $F(a)$ has all coordinates nonzero: $prod f_i$ then cuts out $X$, but since any polynomial cutting out $X$ has degree at least $n$, the conclusion follows.

More generally, for a variety $X subseteq k^n$, define $C(X)$ to be the minimum of the sum of the degrees of the coordinate functions over all polynomial maps $F$ where $F(X) cap F(k^n – X) = emptyset$. Is this quantity equivalent to something that’s well known? You trivially have $C(X) le n$ by taking $F$ to be the identity map. Also, if $X$ is defined by equations whose degree-sum equals $m$, $C(X) le m$. Is one of these inequalities always sharp?