Parity-triviality conjecture for special modules

Let $V$ be a special module in $\mathcal{T}_n^+$, meaning that $V$ is indecomposable, $\operatorname{sdim}(V)=1$, $V^*\cong V$, and $H^0(V)$ contains the trivial module. Parity-triviality conjecture. Up to a parity shift,

$$V\cong {\bf 1}.$$

This is a parity-refined formulation of the triviality conjecture for special modules and would control the possible one-dimensional objects in the category.