F-trivialization conjecture for complete local rings

Let $R$ be a complete $d$-dimensional local ring of positive characteristic, and let $M$ be a $d$-dimensional $R$-module. Write $\mathbf{F}_{*}^nM$ for the $n$-fold Frobenius transform of $M$. The module $M$ is decomposable if it is not indecomposable. F-trivialization conjecture. For every $d$-dimensional $R$-module $M$, there exists some $n$ such that $\mathbf{F}_{*}^nM$ is decomposable. The paper formulates this weaker Frobenius-trivialization property in top dimension because that is the case needed for its results, and suggests an analogous statement for any non-simple module; no resolution is supplied.