Triviality conjecture for special modules

Let $\mathcal{T}_n^+$ be the relevant tensor category. An indecomposable module $V$ in $\mathcal{T}_n^+$ is special if $\operatorname{sdim}(V)=1$, $V^*\cong V$, and $H^0(V)$ contains the trivial object ${\bf 1}$. Triviality conjecture for special modules. Every special module is trivial:

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

This conjecture would rule out additional one-dimensional representations arising from special modules and would help determine the character groups of the associated Tannakian groups.