Four-point map-counting lower-bound conjecture

Let $\mathcal{M}_{0,4}(r_1,r_2,r_3,r_4)$ be the set of maps with the specified vertex data, and order signatures componentwise: $\sigma ot greater than or equal to\sigma'$ when $\sigma_x ot greater than or equal to\sigma'_x$ for some $x otin{ s,t,u}$. Let $\mathcal{M}_{0,4}(r_1,r_2,r_3,r_4\mid\sigma_s,\sigma_t,\sigma_u)$ denote the maps whose signature is at least $\sigma=(\sigma_s,\sigma_t,\sigma_u)$, and let $\mathcal{M}^c_{0,4}(r_1,r_2,r_3,r_4)$ be the weakly connected maps. Four-point map-counting lower-bound conjecture. For any $r_1,r_2,r_3,r_4$ and $\sigma ot greater than or equal to(\frac12,\frac12,\frac12)$, one has

$$\left|\mathcal{M}_{0,4}(r_1,r_2,r_3,r_4\mid\sigma)\right| \geq \left|\mathcal{M}^c_{0,4}(r_1,r_2,r_3,r_4)\right| - \sum_{x\in\{s,t,u\}} \left\lfloor \left(\sigma_x -\tfrac12\right)^2\right\rfloor.$$

The claim is presented as a lower bound motivated by the conformal-bootstrap interpretation of sets of maps with a minimum signature; its resolution is not specified in the supplied text.