Thompson's conjecture on the monster group's graded representation

Let $\tilde J=J-744$ be the normalised $J$-function, and let $\rho_ atural$ be a graded representation on an infinite-dimensional vector space

$$V^ atural=\bigoplus_{i\geq-1}V^ atural_i$$

whose graded parts $V^ atural_i$ are finite dimensional. Thompson's conjecture. There is a somehow natural infinite-dimensional graded representation of the monster group $d4dc$ with

$$\tilde J(\tau)=\sum_{i\geq-1}\dim(V^ atural_i)q^i.$$

Equivalently, the elements of $d4dc$ act naturally as infinite block-diagonal matrices, with every block of finite size, on an infinite-dimensional graded vector space, whose graded dimension is the $q$-expansion of the normalised $J$-function. This conjecture proposes that the coefficients of the normalised $J$-function arise as graded dimensions of a natural monster-group representation; it is the first representation-theoretic formulation of Monstrous Moonshine, and the supplied text gives no resolution status.