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

Let $S_+$ be the subset of positive modes in the twisted mode expansion

$$Y_\sigma(\nu^d,z)=\sum_{k\in\mathbb{Z}}W^d_kz^{-k-d},\qquad Y_\sigma(\widetilde{\nu}^{r+1},z)=\sum_{k\in\frac12+\mathbb{Z}}\widetilde{W}^{r+1}_kz^{-k-r-1}.$$

Set $S'=\{\widetilde{\mathsf{W}}^{r+1}_{1/2}\}\in S_+$. Type C extraneous-mode conjecture. The subset $S'$ is extraneous according to the source's definition. This property is required before constructing the Airy-structure differential representation for the positive modes; the source explicitly says that the available proposition does not prove it in this case.