In this talk, I will introduce a conjecture of Prasad classifying distinguished representations with respect to a quadratic character in terms of certain functorial property of their associated Langlands parameters when the corresponding homogeneous space is a Galois symmetric space. I will also talk about recent progress towards this conjecture and our work on this conjecture for regular supercuspidal representations based on the deep machinery developed by Kaletha and Hakim-Murnaghan.