Generalized moonshine conjecture for the monster

Let $\mathbb{M}$ be the monster, let $g,h\in\mathbb{M}$ be commuting elements, and let $\mathfrak{H}$ denote the upper half-plane. A generalized character is a function $Z$ associating a holomorphic function on $\mathfrak{H}$ to each commuting pair $(g,h)$. Generalized moonshine conjecture. There exists a generalized character $Z$ satisfying the following conditions: (1) $Z(g,h,\tau)$ is invariant under simultaneous conjugation of $g$ and $h$; (2) for any $\binom{a\ b}{c\ d}\in SL_2(\mathbb{Z})$, $Z(g^ah^c,g^bh^d,\tau)=\gamma Z(g,h,(a\tau+b)/(c\tau+d))$ for some constant $\gamma$ depending on the matrix; (3) for fixed $g$, the coefficients of the $q$-expansion of $Z(g,h,\tau)$ form characters of a graded representation of a central extension of $C_{\mathbb{M}}(g)$; (4) $Z(g,h,\tau)$ is either constant or a genus-zero function; and (5) $Z(g,h,\tau)=J(\tau)$ for all $\tau\in\mathfrak{H}$ if and only if $g=h=1$. This conjecture extends monstrous moonshine from the identity-twisted setting to commuting pairs in the monster. It remains open, although Borcherds's proof of the original moonshine conjecture establishes the genus-zero assertion for $Z(1,h,\tau)$, and other special cases have been resolved.