Twisted quantum Hamiltonian reduction conjecture for Ramond modules

Let $\widehat{\mathfrak g}^{\rm tw}$ be the twisted affine algebra, let $\mathcal O$ be its category of modules, and let $H_j$ denote the homology of the twisted quantum Hamiltonian reduction complex. Let $\widehat\nu_s$ be a highest weight, let $\ell(s)=A(k,\nu)$ be the exceptional condition appearing in the paper, and let $W^k_{\min}(\mathfrak g)$ be the minimal $W$-algebra. Twisted reduction conjecture.

For every $M\in\mathcal O$, $H_j(M)=0$ for $j\ne0$. If $\widehat\nu_s$ is nondegenerate, so that $H(L(\widehat\nu_s))\ne0$, then if $\theta/2$ is not a root of $\mathfrak g$, $H_0(L(\widehat\nu_s))$ is irreducible; if $\theta/2$ is a root of $\mathfrak g$, $H_0(L(\widehat\nu_s))$ is irreducible or is a direct sum of two irreducible $W^k_{\min}(\mathfrak g)$-modules, and in the latter case $\ell(s)=A(k,\nu)$ and the second module is isomorphic to the first with opposite parity. This is the twisted analogue of the resolved irreducibility result for ordinary quantum Hamiltonian reduction and is needed for the paper's character and unitarity arguments.