Distinctness of products of Fibonacci numbers

Question

Suppose that $S$ and $T$ are sets of Fibonacci numbers greater than $1$. Let $S^*$ be the product of numbers in $S$, and likewise for $T^*$. If $S^*=T^*$, must $S=T$?

Answer 1

(Finite sets, of course, if you don't want to allow $S^* = T^* = \infty$)

Yes. We may assume wlog $S$ and $T$ are disjoint, and $\max S < \max T$. By Carmichael's theorem, every Fibonacci number except $1$, $8$ and $144$ has a prime factor that does not divide any earlier Fibonacci number. So $\max T$ can only be $8$ or $144$, and neither of these work.