Multivariate extension of Theorem 3 for distributions with densities

Let $F$ be a multivariate distribution admitting a density, with the relevant multivariate iteration map and a suitably reformulated version of Theorem 3. Multivariate extension conjecture. Suitably reformulated, Theorem 3 is valid for any multivariate distribution admitting a density. This extends the preceding bivariate conjecture to the multivariate setting, but the source does not specify the reformulation or establish the claim.