A counterexample to symmetry of $L^p$ norms of eigenfunctions

We answer a question of Jakobson and Nadirashvili on the asymptotic behavior of the $L^p$ norms of positive and negative parts of eigenfunctions of the Laplacian. More precisely, we show that there exists a sequence of eigenfunctions ${\psi }_n$ on the flat $d$-torus for $d\ge 3$, with eigenvalues ${\lambda }_n\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$, such that the ratio $\mathrm{\parallel }{\psi }_n{\chi }_{\{{\psi }_n>0\}}{\mathrm{\parallel }}_p\mathrm{/}\mathrm{\parallel }{\psi }_n{\chi }_{\{{\psi }_n<0\}}{\mathrm{\parallel }}_p$ does not tend to $1$ as $n\to \mathrm{\infty }$ for $1<p\le \mathrm{\infty }$. Our argument is elementary and computer-assisted.