Conway-Norton's Monstrous Moonshine conjecture

Let $d4dc$ be the monster group, let

$$V^ atural=\bigoplus_iV^ atural_i$$

be an infinite-dimensional graded representation with finite-dimensional graded parts, and for each conjugacy class $[g]$ of $d4dc$ let

$$T_{[g]}=\sum_{i\geq-1}\operatorname{Tr}(\rho_\mathbb{M}(g)_{|V^ atural_i})q^i$$

be the associated McKay–Thompson series. Conway-Norton's Monstrous Moonshine conjecture. For every conjugacy class $[g]$ of the monster, $T_{[g]}$ is the $q$-expansion of the normalised Hauptmodul of some subgroup $d4a2_{[g]}$ of $\operatorname{PSL}_2(\mathbb{R})$ commensurable with $\operatorname{PSL}_2(\mathbb{Z})$. This extends Thompson's conjecture by requiring the corresponding trace series for all monster conjugacy classes to be normalized Hauptmoduln. The supplied text attributes the extension to Conway and Norton and gives no resolution status.