A new conjecture

I find a new conjecture about the period of fibonacci sequences modulo m that is the first zero term of fibonacci sequences modulo m giving a similar property.

This conjecture is basis on the calculation of Maple showing that the value is true for all pre-20-terms primes (It is difficult to calculate for more then the 20-terms primes.).

I think this conjecture is quite easy, but I'm too lazy to prove now. However, I think the original conjecture of fibonacci sequences modulo m was equivalent to prove the relation between the first zero term of fibonacci sequences modulo m and m-square.