Existence conjecture for monopoles with prescribed magnetic and holomorphic charges

Fix a nonzero mass $\mu\in\mathfrak g$. Choose a maximal torus and the associated simple roots so that the charges are divided into nonnegative magnetic charges $(m_1,\ldots,m_s)$ and holomorphic charges $(h_1,\ldots,h_{r-s})$. The holomorphic charges are required to satisfy

$$\sum_{l=1}^{r-s} \frac{2\langle\alpha_i,\alpha_{l+s}\rangle}{\langle\alpha_{l+s},\alpha_{l+s}\rangle}h_l+\sum_{j=1}^s\frac{2\langle\alpha_i,\alpha_j\rangle}{\langle\alpha_j,\alpha_j\rangle}m_j\le 0, \qquad i=s+1,\ldots,r.$$

Charge-existence conjecture. For a given $\mu$ there are monopoles with any collection of nonnegative magnetic charges $(m_1,\ldots,m_s)$. Given a choice of magnetic charges, there are monopoles with any collection of holomorphic charges $(h_1,\ldots,h_{r-s})$ satisfying the displayed constraint. This predicts existence in every charge sector allowed by the root-theoretic inequalities, extending beyond the cases where existence was known.