Extraneous-mode conjecture for the type B twisted W-algebra module

Let $r$ be a positive integer, let $\epsilon(i)$ denote the twist shift for the strong generators, and define the positive modes

$$S_+ = \{\mathsf{W}^i_k\mid i\in [2r],\ k\in(\mathbb{Z}_{>0}-\epsilon(i))\}.$$

Set $S'=\{\mathsf{W}^{2r-1}_{1/2}\}\subset S_+$. Type B extraneous-mode conjecture. The subset $S'$ is extraneous in the sense of the source's definition. This condition is needed to construct the shifted Airy structure whose partition function gives the desired Whittaker vector; the source states that the relevant subalgebra condition is known, but that existence of this extraneous subset has not been proved for the twisted module.