Spectral-flow and quantum-Hamiltonian-reduction conjecture for admissible modules

Let $\mathrm{Pr}_{\mathbb{Z}}^k$ be the specified set of integral principal admissible weights, let $S_\theta$ denote spectral flow, let $\mathbf{L}_k(\lambda)$ be the corresponding $\mathcal{W}_k^{D^+}(n,1)$-module, and let $H_f^0(L_k(\mu))$ denote quantum Hamiltonian reduction. Let $\varsigma$ be the involution determined by

$$S_{\bar q}\mathbf{L}_k(\lambda)\simeq\mathbf{L}_k(\varsigma\circ\lambda).$$

Spectral-flow conjecture. For every $\lambda\in\mathrm{Pr}_{\mathbb{Z}}^k$ and $0\leq\theta\leq\bar q$, there exists $\mu\in\mathfrak{h}^*$ such that

$$S_\theta\mathbf{L}_k(\lambda)\simeq H_f^0(L_k(\mu)).$$

Moreover, $\varsigma=\sigma$, equivalently

$$S_{\bar q}\mathbf{L}_k(\lambda)\simeq\mathbf{L}_k(\sigma\circ\lambda).$$

The surrounding results establish periodicity and the existence of an involution, while the asserted identification with $\sigma$ and the reduction description are left as conjectural in the source.