Zeros of characters

Question

I was wondering if the following is true:

Let $\chi$ and $\chi'$ be two irreducible characters over $\mathbb{C}$ of a finite group with same degree.
Suppose that $\chi(g) = 0 \implies \chi'(g) = 0$.
Is true that $\chi'(g) = 0 \implies \chi(g) = 0$?

In other words, can the zero set of a character be a strict subset of the zero set of another character of the same degree?

Any counter-example or help proving this is appreciated.

Answer 1

Check out the dihedral group $D_{12}$.