I thought I would add my thoughts.
Using the constraints for moduluses in this question, one can find the periods of the fibbonacci sequence modulus the same integers:
$F_n \mod 7$ gives the following period
$F_n \mod 9$ gives the following period
$F_n \mod 63$ gives the following period
Using these numbers one can derive:
if $F_n = a^3+b^3$, none of the following have an integer solution.
So if one can show that all integers above some constant satisfy atleast one of the above, and then test $F_1...F_c$ for having being a sum of 2 cubes where $c$ is that constant, you would have a proof.