Benson's conjecture on tensor products of odd-dimensional representations

Let $G$) be a finite $2$-group, let $\Bbbk$ be a field of characteristic $2$, and let $V$ be an odd-dimensional representation of $G$. An indecomposable representation is called a $2'$-representation when its dimension is not divisible by $2$, and it is called $2'$-invertible when its tensor product with its dual decomposes as the direct sum of the trivial representation and representations whose dimensions are divisible by $2$.

Benson's conjecture. All indecomposable summands of $V\otimes V^*$ have dimension divisible by $2$, except for the single summand isomorphic to the trivial representation $\Bbbk$; equivalently, every $2'$-representation is $2'$-invertible.

This conjecture is attributed to Dave Benson and is based on extensive computer-algebra evidence. The paper notes a stronger version in which all nontrivial summands have dimension divisible by $4$, but the supplied text does not state its resolution.