Shimura operators for certain Hermitian symmetric superpairs

We give a partial super analog of a result obtained by S. Sahi and G. Zhang relating Shimura operators and certain interpolation symmetric polynomials. In particular, we study the pair $(\mathfrak{gl}(2p|2q), \mathfrak{gl}(p|q)\oplus\mathfrak{gl}(p|q))$, define the super Shimura operators in $\mathfrak{U}(\mathfrak{g})^\mathfrak{k}$, and using a new method, prove that their images under the Harish-Chandra homomorphism are proportional to Sergeev and Veselov's Type $BC$ interpolation supersymmetric polynomials, under the assumption that a family of irreducible $\mathfrak{g}$-modules are spherical. We prove this conjecture using the notion of quasi-sphericity for Kac modules when $p=q=1$, and give explicit coordinates of (quasi-)spherical vectors.