Failure of $L^r$-Calderón-Zygmund estimates for the p-Laplace equation for small $r$

Let $p\mathrm{\ne }2$. For any small enough $r>\max \{p-1,1\}$ and for any $\mathrm{\Lambda }>1$ there exists a Lipschitz function $u$ and a bounded vectorfield $f$ such that $$\{\begin{array}{ll}\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=\mathrm{d}\mathrm{i}\mathrm{v}(f)& \text{in }{\mathbb{B}}^2\\ u=0& \text{on }\mathrm{\partial }{\mathbb{B}}^2\end{array}$$ but $${\int }_{{\mathbb{B}}^2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^r\le ̸\mathrm{\Lambda }{\int }_{{\mathbb{B}}^2}\mathrm{\mid }f{\mathrm{\mid }}^{\frac{r}{p-1}}\mathrm{.}$$ This disproves a conjecture by Iwaniec from 1983. The proof adapts recent convex-integration ideas by Colombo-Tione.