Most likely position for random walk with symmetric jumps after $t$ steps

Question

Consider a random walk on $\mathbb{Z}$ starting at 0 with jump distribution $p(x)$ such that,

1. $p(x) = p(-x)$

2. $p(x)>0$ for all $x \in \mathbb{Z}$

Let $p^{\,t}(n)$ be the probability that the random walk is at $n \in \mathbb{Z}$ at time $t \in \mathbb{N}$. How to prove that, for any $t$ even and large enough, $$ \max\{ p^t(n) \, ; \, n \in \mathbb{Z}\} $$ is attained at $n=0$?

Comment 1: I probably need a proper local central limit theorem.

Comment 2: I am also trying to prove it by employing characteristic functions.