Infinite transitivity conjecture for homogeneous quasi-affine varieties

Let $X$ be an irreducible homogeneous quasi-affine variety with

$$\dim X\geq 2$$

and with $\mathbb{K}[X]^{\times}=\mathbb{K}^{\times}$. Here $\theta(X)$ denotes the transitivity degree of the automorphism group of $X$. Infinite transitivity conjecture. Then

$$\theta(X)=\infty.$$

This conjecture asserts that, under the stated mild necessary conditions, homogeneity forces the automorphism group to act infinitely transitively. The surrounding results establish the claim for several classes of homogeneous varieties, but the general statement remains open.