Some conjectures regarding certain Schubert structure constants in Lie types B and D

I give the details of some conjectures regarding Schubert calculus in Lie types B and D. Specifically, I conjecture rules for Schubert structure constants $c_{u,v}^w$ when $X_{w_{0}u}^v$ is a Richardson variety stable under the spherical Levi subgroup ${\text{C}}^{\ast }\times SO(n-2,\text{C})$ of $SO(n,\text{C})$.