ELMO 2020 Problem 2

Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$ . Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$ . Must $k$ be a Fibonacci number?

Proposed by Fedir Yudin.