This is related to my question, but it concerns a specific point of the proof of Schur's Theorem.

Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that $$\forall g\in G,\qquad\sum_{h\in G}\overline{\chi(h)}\chi(gh)=\frac{|G|}{\chi(e)}\,\chi(g)\quad?$$ This does not seem to be related to the orthogonality properties of the table of characters. This obviously true if $\chi$ is a linear character, or if $g=e$. I checked it for $G=\frak S_3$ and for one nonlinear character of $\frak S_4$.