Kudla Millson lift of toric cycles and restriction of Hilbert modular forms

Let $V$ be quadratic space of even dimension and of signature $(p, q)$ with $p \geq q > 0$. We show that the Kudla-Millson lift of toric cycles - attached to algebraic tori - is a cusp form that is the diagonal restriction of a Hilbert modular form of parallel weight one. We deduce a formula relating the dimension of the span of such diagonal restrictions and the dimension of the span of toric and special cycles.
Comment: v2: References added v3: We remove the unnecessary anisotropy condition for the quadratic space in section 5, that rendered Corollary 1.2.1 empty