Sums of subsequences of the Fibonacci numbers

Question

I read an interesting question on here yesterday Sums of subsequences of Tribonacci number. It turns out that if you have two subsequences of the Tribonacci numbers that have the same sum, you can't in general shift the indices and keep the equality invariant. But I was wondering, since there are more "small" numbers in the Fibonacci sequence, does the same thing hold for the Fibonacci numbers. I wrote a quick program to check if there were any small counterexamples, but it seems like it is true. I don't see a way to prove it prove it directly. There might be some properties of the Fibonacci sequence one needs to prove first. Any help is appreciated.