In a Markov Decision Process (MDP), the discounted total reward is defined as $\sum_{t=0}^\infty \gamma^tr_t$ where $r_t$ is the reward perceived at time $t$ and $\gamma$ is a real number $\in ]0, 1[$. The average total reward is defined as $\lim_{t\rightarrow \infty}\frac{\sum_{i=0}^tr_i}{t}$.

My question is: does a policy $\pi$ that maximizes the discounted total reward also maximize the average total reward and vice versa? Or there is a policy $\pi$ that maximizes the first and $\pi'$ that maximizes the second where $\pi \neq \pi'$?