A nowhere dense space of linear superpositions of functions of several variables

Let $\mathbf I^3$ be the unit cube of three-dimensional space $R^3$, and let $\Phi_i(x)$, $i=1,\dots,n$, be mappings $\Phi_i\colon\mathbf I^3\to R^2$ of class $C_2$. We prove that the set of functions $F(x)$ on $\mathbf I^3$ which can be represented in the form
$$ F(x)=\sum_{i=1}^n(\chi_i\circ\Phi_i)(x), $$
where the $\chi_i(u_1,u_2)$ are arbitrary continuous functions, $\chi_i\colon R^2\to R$, is nowhere dense in $\mathscr L_2(\mathbf I^3)$.