The linear independence conjecture for Poincaré series of Mitchell–Priddy summands

Let $L_\lambda$ be the indecomposable injective unstable module indexed by a partition $\lambda$, and let $P_\lambda(q)$ denote its Poincaré series. Linear independence conjecture. The Poincaré series $P_\lambda$ are linearly independent. This conjecture is refuted: an explicit nonzero virtual module has trivial Poincaré series, although its image under Lannes' $T$-functor has nontrivial Poincaré series.