The torus-amplitude conjecture for logarithmic minimal models

Let $p_+$ and $p_-$ be the parameters of the logarithmic $(p_+,p_-)$ model. Let ${\bar{\mathscr{C}}}$ be the space of generalized characters, let $\mathscr{G}$ be the subspace described in the preceding decomposition, and let $\mathscr{C}$ denote the space of torus amplitudes. The modular group $SL(2,\mathbb{Z})$ acts on these spaces. Torus-amplitude conjecture. The $SL(2,\mathbb{Z})$-representation generated from $\mathscr{G}$ coincides with the space of torus amplitudes, namely

$${\bar{\mathscr{C}}}=\mathscr{C}$$

as $SL(2,\mathbb{Z})$ representations. This conjecture identifies the generalized-character space with the torus-amplitude space and is proposed as a highly probable description of the field content of a consistent logarithmic conformal field theory model. The supplied passage gives no resolution, so the conjecture remains open.