Generalized sign Fourier uncertainty

We consider a generalized version of the sign uncertainty principle for the Fourier transform, first proposed by Bourgain, Clozel and Kahane in 2010 and revisited by Cohn and Gon\c{c}alves in 2019. In our setup, the signs of a function and its Fourier transform resonate with a generic given function $P$ outside of a ball. One essentially wants to know if and how soon this resonance can happen, when facing a suitable competing weighted integral condition. The original version of the problem corresponds to the case $P \equiv 1$. Surprisingly, even in such a rough setup, we are able to identify sharp constants in some cases.