Triviality conjecture for the one-dimensional module $I$

Let $I$ be the indecomposable module described in the source: it has superdimension $1$, is self-dual under both dualities, occurs as a multiplicity-one summand of $L\otimes L^\vee$, and satisfies the stated cohomological and negligible-summand conditions. Triviality conjecture for $I$.

$$I\simeq {\mathbf{1}}.$$

This is a special case intended to imply the conjecture that all special modules are trivial, but the source states that it is currently unproved.