Character-valuedness of the local conjugating class function

Prove that for every finite group G and every collection of primes π, the class function Π_{π,G}, defined by Π_{π,G}(g)=|C_G(g)| for π-elements g and Π_{π,G}(g)=0 otherwise, is a character of G; equivalently, establish that [Π_{π,G},χ]≥0 for every irreducible character χ of G.

Background

The paper defines the π-local conjugating class function Π{π,G} by assigning the centralizer order |C_G(g)| to each π-element of a finite group G and assigning zero to all other elements. The authors prove that Π{π,G} is always a generalized character, and they prove that it is an ordinary character when π consists of the p-regular elements for a prime p or when G is π-separable.

The conjecture seeks to remove both of those restrictions and assert character-valuedness for arbitrary finite groups and arbitrary collections of primes. The authors report computational evidence supporting the claim and note that it would answer a question posed by Robinson. They also show that a potential counterexample cannot arise from the Weil character of degree q^2−q for PSU_3(q) when (3,q+1)=1.