Moonshine-like function characterization conjecture

Let $f$ be a moonshine-like function, meaning a completely replicable function given by the graded trace of a finite-order automorphism of a $C_2$-cofinite, holomorphic vertex operator algebra of central charge $24$ with graded dimension $J(\tau)=j(\tau)-744=q^{-1}+196884q+\cdots$. Moonshine-like function characterization conjecture. There is some $g\in\mathbb{M}$ such that

$$f=T_g.$$

This conjecture would characterize all trace functions arising from moonshine-like vertex operator algebras as monstrous McKay–Thompson series. The paper proves strong restrictions and shows that all non-monstrous functions except possibly $9a$, $63a$, and $117a$ are non-moonshine-like, leaving only a small unresolved part.