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

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 $m$, the conclusion follows.

I am particularly interested in the case when $\deg(f_i) = 2$.