  The Shimura operators are a certain distinguished basis for invariant
differential operators on a Hermitian symmetric space. Answering a question of
Shimura, Sahi--Zhang showed that the Harish-Chandra images of these operators
are specializations of certain $BC$-symmetric interpolation polynomials that
were defined by Okounkov.
  We consider the analogs of Shimura operators for the Hermitian symmetric
superpair $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}=
\mathfrak{gl}(2p|2q)$ and $\mathfrak{k}= \mathfrak{gl}(p|q)\oplus
\mathfrak{gl}(p|q)$ and we prove their Harish-Chandra images are
specializations of certain $BC$-supersymmetric interpolation polynomials
introduced by Sergeev--Veselov.
