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 Distinctness of products of Fibonacci numbers, in which $S$ and $T$ are sets of Fibonacci numbers, for which the answer is "yes", as a corollary of Carmichael's theorem.

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 Distinctness of products of Fibonacci numbers, in which $S$ and $T$ are sets of Fibonacci numbers, for which the answer is "yes", as a corollary of Carmichael's theorem.