Number of continuous characters on an infinite Hausdorff precompact abelian group with exponent $p$

Let $(G,\mathcal T)$ be an infinite Hausdorff precompact abelian topological group and let $G$ have exponent $p$ where $p$ is a prime number.

Can it be proved that there are at least $p+1$ continuous homomorphisms $f:G\to \Bbb T$, where $\Bbb T$ is the circle group?

By Proposition 3.4 of

D. Dikranjan; L. Stoyanov. An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups. Topology Appl. 158 (2011), no. 15, 1942--1961.

if $G$ is an infinite abelian group and $\mathcal T$ is a Hausdorff precompact group topology on $G$, there are infinitely many continuous homomorphisms $f:G\to \Bbb T$.