Equivalence of the four transitivity conditions for quasiaffine varieties

Let $X$ be an irreducible quasiaffine variety of dimension at least $2$. Let conditions (1)–(4) refer to the four conditions in Theorem~, whose equivalence is being conjectured. Transitivity-conditions conjecture. Conditions (1)–(4) of Theorem~ are equivalent for $X$. The paper presents this as a conjectural completion of the implications proved under additional hypotheses, so the equivalence remains open in the stated generality.