Irreducibility conjecture for Wakimoto modules with Whittaker Heisenberg factors

Let $\frak{g}$ be the Lie algebra under consideration, let $\frak{h}$ be its Cartan subalgebra, and let $u$ be a non-critical invariant bilinear form with corresponding affine level $u$. Let $W_{\frak{g}}$ be the Weyl vertex algebra associated with the negative nilpotent part of $\frak{g}$, and let $N_1(\bm{\lambda})$ be a Whittaker, non-highest-weight module for the Heisenberg vertex algebra $\pi^{k+h^{\vee}}$. Via the Frenkel–Feigin homomorphism, regard $W_{\frak{g}}\otimes N_1(\bm{\lambda})$ as a module over the universal affine vertex algebra $V^{\kappa}(\frak{g})$. The Wakimoto irreducibility conjecture. If $\kappa$ is non-critical, then

$$W_{\frak{g}} \otimes N_1(\bm{\lambda})$$

is an irreducible $V^{\kappa}(\frak{g})$-module. This extends the known generic highest-weight irreducibility results to non-highest-weight Whittaker modules; determining irreducibility at non-critical levels is the main question proposed here, while the critical-level analogues are connected with the Kac–Kazhdan conjecture.