  In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption
that either $p^2 \nmid \chi(1)$ for all $\chi \in {\rm Irr}(G)$ or $p^2 \nmid
\phi(1)$ for all $\phi \in {\rm IBr}_p(G)$. This would settle two conjectures
raised by Lewis, Navarro, Tiep, and Tong-Viet.
