Lie powers of modules for cyclic p-groups

We consider the decomposition problem for Lie powers of finite-dimensional modules for a cyclic p-group C over a field K of prime characteristic p. That is, given a finite-dimensional KC-module V and a positive integer n we would like to be able to decompose the n-th Lie power $L^n(V)$ as a direct sum of indecomposable KC-modules, describing which isomorphism types of indecomposable KC-modules occur in such a decomposition and with what multiplicity. By a theorem of R. M. Bryant and M. Schocker the problem reduces to the case $n= p^m$, for $m \geq 1$. In this talk I will discuss some conjectured recursive descriptions of such Lie powers up to isomorphism.