Inequalities related to the Pisano Period

Let $$(F_n)_n$$ be the Fibonacci sequence and $$\pi(m)$$ the well-known Pisano period of $$m$$ (i.e., the smallest period of $$F_n\pmod m$$).

The conjecture is that $$m$$ divides $$\pi(m^2)$$ for all $$m\geq 1$$ which yields the upper bound $$\pi(m^2)\geq m$$ (which is sharp). However, I am interested in the following much wearker inequality: $$\pi(m^2)>\sqrt{m}$$, for all $$m\geq 1$$.

Clearly, it is easy to prove that $$\pi(m^2)>c\log m$$ (by definition). But how to get a better lower bound for $$\pi(m^2)$$ (something of $$O(m^{\epsilon})$$, for $$\epsilon\in (0,1)$$)?

Someone can help me?