The affine-space parametrization conjecture for complements of codimension-two subvarieties

Let $Z\subset \mathbb{A}^n$ be an algebraic subvariety of codimension at least $2$. Suppose $W_1,\ldots,W_N\subset \mathbb{A}^n$ are hypersurfaces containing $Z$ with

$$\bigcap_i W_i=Z,$$

and let $G_i\subset \operatorname{Aut}(\mathbb{A}^n,W_i)$ be $\mathbb{G}_a$-subgroups. Choose a point $p\in \mathbb{A}^n\setminus Z$ and a subvariety $H\subset \mathbb{A}^N$ isomorphic to $\mathbb{A}^n$. Define

$$\psi=(G_N\cdot\ldots\cdot G_1).p\colon \mathbb{A}^N\longrightarrow \mathbb{A}^n\setminus Z.$$

Conjectural approach. There exist such $N$, hypersurfaces $W_i$, $\mathbb{G}_a$-subgroups $G_i$, a point $p$, and a subvariety $H$ for which the restricted map $\psi|_H$ is surjective. This would provide a surjective morphism from an affine space isomorphic to $\mathbb{A}^n$ onto $\mathbb{A}^n\setminus Z$.