  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.
