The spectral fractional-Laplacian sign-transform inequality conjecture

Let $s\in(1,\frac32)$, and let $u\in \widetilde H^s(\Omega)$ be sign-changing. The spectral quadratic forms associated with the fractional Laplacian should satisfy inequalities analogous to

$$Q_s^{\rm DR}[u]<Q_s^{\rm DR}[|u|].$$

Spectral sign-transform inequality conjecture. For $s\in(1,\frac32)$, the analogous inequalities should hold for spectral quadratic forms. The corresponding inequality has been proved for the restricted (Dirichlet realization) fractional Laplacian, while the spectral case is presented as an open question.