The vanishing conjecture for Loewy-length-three summands

Let $E=(\mathbb{Z}/p)^2$ and let $kE$ be its group algebra. For a $kE$-module $M$ of constant Jordan type, let $\mathfrak{K}$ and $\mathfrak{I}$ denote the functors used to form the subquotients $\mathfrak{K}^2(M)/\mathfrak{I}^2\mathfrak{K}^2(M)$ and $\mathfrak{K}^2(M/\mathfrak{I}^2(M))$. Let $N$ be an indecomposable direct summand of either subquotient, and suppose that $N$ has Loewy length three. The vanishing conjecture. The associated sheaf $\mathcal{F}_1(N)$ is the zero sheaf on $\mathbb{P}^1(k)$. This conjecture predicts that any possible Loewy-length-three summands in these subquotients make no contribution to the vector bundle invariant $\mathcal{F}_1(M)$; the source provides no resolution of the claim.