Weakly connected maps conjecture for non-diagonal conformal-bootstrap spectra

Consider an $n$-point function of diagonal and non-diagonal fields on a Riemann surface of genus $g$. Let $\mathcal{M}^c_{g,n}(r_i)$ denote the set of weakly connected maps with the corresponding external first Kac indices $r_i$, and let the conformal-bootstrap spectra contain only non-diagonal fields. Write $d_{0,4}^c(\delta_i)$ for the dimension of the space of four-point crossing-symmetry solutions. Weakly connected maps conjecture. For any such $n$-point function, the dimension of the conformal-bootstrap solution space equals $|\mathcal{M}^c_{g,n}(r_i)|$; in particular,

$$d_{0,4}^c(\delta_i)=\left|\mathcal{M}^c_{0,4}(r_i)\right|.$$

This conjecture motivates the definition of weakly connected maps, which can have vertices of valency zero and therefore account for correlation functions involving diagonal fields. The supplied text gives no resolution status.