Bounding the number of p'-degrees from below

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in {\text{Syl}}_p(G)$. We prove a recent conjecture by Hung stating that $\mathrm{\mid }{\text{Irr}}_{p^{\mathrm{\prime }}}(G)\mathrm{\mid }\ge \frac{\exp (P\mathrm{/}{P}^{\mathrm{\prime }})-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\ge 2$ be an integer and suppose that $p^a$ does not exceed the exponent of the center of $P$. We then also show that the number of conjugacy classes of elements of $G$ for which $p^a$ is the exact $p$-part of their order is at least $p^{a-1}$.