Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains

Let $u$ be either a second eigenfunction of the fractional $p$-Laplacian or a least energy nodal solution of the equation $(-\Delta)^s_p u = f(u)$ with superhomogeneous and subcritical nonlinearity $f$, in a bounded open set $\Omega$ and under the nonlocal zero Dirichlet conditions. Assuming only that $\Omega$ is Steiner symmetric, we show that the supports of positive and negative parts of $u$ touch $\partial\Omega$. As a consequence, the nodal set of $u$ has the same property whenever $\Omega$ is connected. The proof is based on the analysis of equality cases in certain polarization inequalities involving positive and negative parts of $u$, and on alternative characterizations of second eigenfunctions and least energy nodal solutions.