Finite-generation conjecture for infinite transitivity on generically flexible varieties

A family $\mathfrak{F}$ of $\mathbb{G}_a$-subgroups of $\operatorname{Aut}(X)$ is called saturated if it contains every replica of each of its members and is closed under conjugation by the subgroup it generates. A variety $X$ is generically flexible if $\operatorname{SAut}(X)$ acts on $X$ with an open orbit. For a finite collection of $\mathbb{G}_a$-subgroups $H_1,\ldots,H_N$ of $\operatorname{Aut}(X)$, write $G=\langle H_1,\ldots,H_N\rangle$. Finite-generation conjecture. Any generically flexible affine variety $X$ admits a finite collection $\{H_1,\ldots,H_N\}$ of $\mathbb{G}_a$-subgroups of $\operatorname{Aut}(X)$ such that $G$ acts infinitely transitively on its open orbit. The paper develops weaker conditions than saturation that guarantee infinite transitivity and proves that every generically flexible variety has a countable, rather than necessarily finite, generating family; the finite-generation assertion is presented as a conjecture.