Extension of Theorem 3 to bivariate distributions with densities

Let $F$ be a bivariate distribution admitting a density, and let the iteration map and the conclusion of Theorem 3 be understood as in the source paper. Extension conjecture. Theorem 3 is valid for any bivariate distribution admitting a density. The conjecture proposes removing the stronger dependence assumptions used in Theorem 3 while retaining the density hypothesis; whether that hypothesis can also be weakened is left unclear in the discussion.