Simplified variant of Collatz conjecture.

The case of $g_0$ is immediate. You can combine two steps in one and

$$g'_0(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ (n+1)/2 & \text{if } n\equiv 1 \pmod{2}\end{cases}$$ so that the function is strictly decreasing for $n>2$, and $g'_0(1)=g'_0(2)=1$.

The same argument holds for $g_1$ and $g_2$.