The eigenvalue and diagonalisation conjecture for Lannes' T-functor

Let $K^{red}(\mathcal U)$ be the Grothendieck ring of finite direct sums of indecomposable reduced injective unstable modules, and let $K^{red}_n(\mathcal U)$ be the subgroup generated by the classes of the $L_\lambda$ with $\lambda_1\leq n$. Lannes' $T$-functor acts on $K^{red}_n(\mathcal U)$. Eigenvalue and diagonalisation conjecture. The eigenvalues of $T$ on $K^{red}_n$ are all powers of $2$, more precisely they are $(1)^{2^{n-1}},(2)^{2^{n-2}},\ldots,(2^{n-1})^1,(2^n)^1$, where the exponent indicates multiplicity, and $T$ is diagonalisable on $K^{red}_n$. The conjecture was checked through $n=9$ for $p=2$ and was subsequently proved using the Segal conjecture.