On completeness of local intertwining periods

In this paper we study the problem of explicitly describing the space of invariant linear forms on induced distinguished representations in terms of invariant linear forms on the inducing representation. More precisely, for certain tempered reductive symmetric pairs (G,H) over a local field of characteristic zero, which we call unimodular in this paper, we study under which condition on the inducing representation, the space of H-invariant linear forms on a parabolically induced representation of G is generated by regularized intertwining periods attached to admissible parabolic orbits in G{H, as defined in the work of Matringe--Offen--Yang. We conjecture that it is the case when the inducing representation is square-integrable. Under this assumption we actually conjecture that one can replace regularized by normalized intertwining periods. We then verify the conjecture on known examples, and prove it for various pairs where G has semi-simple split rank one.