Suppose that $r$ is a homogeneous linear recurrence sequence of order $>1$ with nonnegative integer coefficients, not all $0$, and nonnegative initial values, not all $0$.  Suppose that $S$ and $T$ are finite sets of numbers in $r$.  Let $S'$ be the product of numbers in $S$, and let $T'$ be the product of numbers in $T$.  If $S' = T'$, must $S = T$?   

This question generalizes https://mathoverflow.net/q/238505, in which $S$ and $T$ are sets of Fibonacci numbers, for which the answer is "yes", as a corollary of Carmichael's theorem.