Rademacher-sum conjecture for umbral McKay–Thompson series

Let $X$ be a Niemeier root system and let $g\in G^X$. Let $\check{H}^X_g$ denote the vector formed by the components of $H^X_g$ indexed by $0<r<m$, and let $R^X_{\Gamma_0(n_g),\check{\nu}^X_g}$ be the associated vector-valued Rademacher sum. For $X=A_8^3$, let $\check{t}^{(9)}_g$ be the vector-valued theta series formed from the components $t^{(9)}_{g,r}$ with $0<r<9$. Rademacher-sum conjecture. If $X\neq A_8^3$, or if $X=A_8^3$ and $g\in G^X$ does not satisfy $o(g)\equiv0\pmod3$, then

$$\check{H}^X_g(\tau)=-2R^X_{\Gamma_0(n_g),\check{\nu}^X_g}(\tau).$$

If $X=A_8^3$ and $g\in G^X$ satisfies $o(g)\equiv0\pmod3$, then

$$\check{H}^X_{g,r}(\tau)=-2R^X_{\Gamma_0(n_g),\check{\nu}^X_g}(\tau)+\check{t}^{(9)}_g(\tau).$$

This replaces the earlier optimal-growth formulation and expresses the umbral McKay–Thompson series through Rademacher sums, with an additional theta-series correction in the exceptional $A_8^3$ case. The supplied text does not state a resolution.