The Mathieu moonshine representation-pair conjecture

Let $K=\bigoplus_{n=-1}^{\infty}K_n q^{n/8}$ be the virtual graded $M_{24}$-module determined by the McKay–Thompson series, with $K_n$ the representation whose character is given by the coefficient of $q^{n/8}$. Let $n=\ell m^2\equiv 7\pmod{8}$. The irreducible representations are denoted by their character labels $\chi_i$, and a pair $(\chi_i,\chi_j)$ means a pair of conjugate irreducible representations. Mathieu moonshine representation-pair conjecture. For $\ell=7$, $K_n$ contains one of the pairs $(\chi_3,\chi_4)$, $(\chi_{12},\chi_{13})$, or $(\chi_{15},\chi_{16})$; for $\ell=15$, it contains the pair $(\chi_5,\chi_6)$; and for $\ell=23$, it contains the pair $(\chi_{10},\chi_{11})$. This conjecture specifies particular irreducible constituents expected in the Mathieu moonshine module from the Fourier coefficients of the McKay–Thompson series. The supplied text does not give evidence that the conjecture has been proved or disproved.