Irreducibility conjecture for Whittaker modules of the symmetric-group orbifold

Let $M(1)$ be the rank-$\ell$ Heisenberg vertex operator algebra, let $S_{\ell}$ act on it by permuting a basis $h_1,\dots,h_{\ell}$ of the underlying space, and let $\boldsymbol{\lambda}:\mathfrak n\to\mathbb C$ be a Whittaker function. Write $M(1,\boldsymbol{\lambda})$ for the corresponding Whittaker module, and define

$$\boldsymbol{\lambda}\circ g=(\lambda^{g(1)},\dots,\lambda^{g(\ell)})$$

for $g\in S_{\ell}$. A $2$-cycle is a transposition in $S_{\ell}$. Irreducibility conjecture. If $\boldsymbol{\lambda}\circ\sigma\ne\boldsymbol{\lambda}$ for every $2$-cycle $\sigma\in S_{\ell}$, then $M(1,\boldsymbol{\lambda})$ is an irreducible module for the fixed-point vertex operator algebra $M(1)^{S_{\ell}}$. The preceding proposition establishes the analogous assertion for $M(1)^{\langle g\rangle}$ for each individual $g\in S_{\ell}$ under the same hypothesis; the conjecture asks for irreducibility under the full symmetric-group orbifold, and no resolution is given here.