The conjecture that all quasihomogeneous varieties are homogeneous

A quasihomogeneous variety is a variety with a group action having an open orbit, while a homogeneous variety has a transitive group action. Homogeneity conjecture. All quasihomogeneous varieties are homogeneous. The claim would identify these two classes of varieties; the source notes that Grassmannians and products provide examples of quasihomogeneous varieties, but does not establish whether they are all homogeneous or give a known counterexample.