Composite-order Tate cohomology conjecture for graded Brauer characters

Let $M$ be the monster group, let $g eq 1$ be an element of order $N$ whose \prime factor is $p$, and let $h e 1$ be an $N$-regular element of the centralizer $C_{M}(g)$. For $i=0,1$, consider the graded $p$-Brauer character

$$\sum_{n\in \mathbb{Z}}\widetilde{\operatorname{Tr}}_p\big(h|\hat{H}^i(g,V_n\otimes \mathbb{Z}_p)\big)q^{n-1}.$$

Composite-order graded Brauer character conjecture. Each of these series is a linear combination of Hauptmoduls. This predicts a uniform Hauptmodul description for the graded Brauer characters arising from Tate cohomology for arbitrary composite-order monster elements.