References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel

Question

I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper

Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^n_a), Int. Math. Res. Not. 1999, No. 22, 1211-1230 (1999). ZBL0966.14033.

The statement following Example 3.6 claimed

1. The moduli spaces of $(A,\mathfrak{m})$, where $\dim A=8+1$ and $\mathfrak{m}^3\ne0,\mathfrak{m}^4=0$, is the moduli space of genus 5 curves;
2. If $\dim A=9+1$ and $\mathfrak{m}^3\ne0,\mathfrak{m}^4=0$, then the moduli space is that of K3 surfaces of degree 8.

The authors thought them interesting as well and asked "Can it be explained geometrically, in terms of birational maps between different $\mathbb{G}_a^n$-structures?"

I want to know:

1. How do these two claims come? Are there references?
2. Are there any studies following Hassett-Tschinkel's question? More correspondence?