The Polydegree Conjecture for plane polynomial automorphisms

Let $\mathcal{G}=\mathcal{G}(\mathbb{C})$ be the group of polynomial automorphisms of the affine plane, and let $\mathcal{G}_{(d_1,\ldots,d_k)}$ denote the set of automorphisms with degree sequence $(d_1,\ldots,d_k)$. For a subset $A\subset\mathcal{G}$, write $\overline{A}$ for its Zariski closure. Let $k<l$, and let $(d_1,\ldots,d_k)$ and $(e_1,\ldots,d_l)$ be two degree sequences satisfying

$$d_1+\cdots+d_k<e_1+\cdots+e_l.$$

Polydegree Conjecture. Then

$$\mathcal{G}_{(d_1,\ldots,d_k)}\subset\overline{\mathcal{G}_{(e_1,\ldots,e_l)}}.$$

This generalizes the known length-two case, where the topological degree constraint is sufficient when the shorter sequence has length one. The corresponding assertion in length three is known to fail in its direct form, while the general containment problem remains open.