The joint Dickman law for consecutive integers

Let $P^{+}(n)$ denote the largest prime factor of n. We prove that $\log P^{+}(n)/\log n$ and $\log P^{+}(n+1)/\log n$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erdős–Pomerance joint Dickman conjecture positively and implies that the ordering $P^{+}(n)<P^{+}(n+1)$ has natural density 1/2.