Infinite transitivity criterion for affine toric surfaces

Let $X$ be an affine toric surface, let $\sigma$ be a cone corresponding to $X$, and let $e_1,\ldots,e_k$ define locally nilpotent derivations $\partial_{e_1},\ldots,\partial_{e_k}$ of its coordinate ring. Set

$$G=\left\langle\exp(\mathbb{K}\partial_{e_1}),\ldots,\exp(\mathbb{K}\partial_{e_k})\right\rangle.$$

Infinite transitivity conjecture for affine toric surfaces. The group $G$ acts on the open orbit of $X$ infinitely transitively if and only if

$$\mathbb{Z}\langle e_1,\ldots,e_k\rangle=\mathbb{Z}\langle\sigma\rangle.$$

This is proposed as a generalization of the paper's results for affine toric surfaces. The source says that the problem can be solved directly by resolving a singular toric surface through a series of equivariant blow-ups and gluing copies of $\mathbb{A}^2$, so the conjecture is treated here as resolved.