  In this paper we answer Iwaniec and Sbordone's conjecture \cite{IB94}
concerning very weak solutions to the $p$-Laplace equation. Namely, on one hand
we show that distributional solutions of the $p$-Laplace equation in $W^{1,r}$
for $p \neq 2$ and $r>\max\{ 1,p-1\}$ are classical weak solutions if their
weak derivatives belong to certain cones. On the other hand, we construct via
convex integration non-energetic distributional solutions if this cone
condition is not met, thus answering negatively Iwaniec and Sbordone's
conjecture in general.
