An obstruction to $\ell^p$-dimension

For any group $G$ containing an infinite elementary amenable subgroup, and any $2<p<\infty$, there exists closed invariant subspaces $E_i\nearrow \ell^pG$ and $F\neq 0$ such that $E_i\cap F = 0$ for all $i$. This is an obstacle to $\ell^p$-dimension and gives a negative answer to a question of Gaboriau.