Normalized character conjecture for the unique Coulomb-branch category-O module

Let $\Phi^{-,\mathrm{re}}_\mu$ be the relevant set of negative real roots, let $\langle\alpha,\mu\rangle$ denote the root–coweight pairing, and let $L$ be the unique irreducible object in the category $\mathcal{O}$ under consideration, with normalized character $\widetilde{\operatorname{ch}}(L)$. Normalized character conjecture. For $\theta>0$,

$$\widetilde{\operatorname{ch}}(L)=\prod_{\alpha\in\Phi^{-,\mathrm{re}}_\mu}\frac{1}{(1-e^{\alpha})^{\langle\alpha,\mu\rangle}}.$$

This gives an explicit character formula conditional on the preceding tangent-space conjecture; the source does not establish it in general.