A conjecture of Cameron and Kiyota on sharp characters with prescribed values

Let $\chi$ be a virtual (generalized) character of a finite group $G$ and $L=L(\chi)$ be the image of $\chi$ on $G-\lbrace 1 \rbrace$. The pair $(G, \chi)$ is said to be sharp of type $L$ if $|G|=\prod _{ l \in L} (\chi(1) - l)$. If the principal character of $G$ is not an irreducible constituent of $\chi$, the pair $(G,\chi)$ is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in $1988$. This conjecture states that if $(G,\chi)$ is sharp and $|L|\geq 2$, then the inner product $(\chi,\chi)_G$ is uniquely determined by $L$. We then prove that this conjecture is true in the case that $(G,\chi)$ is normalized, $\chi$ is a character of $G$, and $L$ contains at least an irrational value.