On the Orbits of Multiplicative Pairs

We characterize all pairs of completely multiplicative functions $f,g:\mathbb{N}\to\mathbb{T}$ such that the orbit closure $$\overline{\{(f(n),g(n+1))\}_{n\ge 1}} \neq \mathbb{T}\times \mathbb{T}.$$ In so doing, we settle an old conjecture of Zolt\'an Dar\'oczy and Imre K\'atai.