Determine the first-order normal-operator subspace for a nonscalar representation in the second unitary symmetric pair

Determine the dimension or precise structure of the subspace of first-order symmetry breaking operators represented by normal derivative differential operators for a nonscalar holomorphic representation of the pair $(\\mathfrak{su}(n,1),\\mathfrak{s}(\\mathfrak{u}(n-1)\\oplus\\mathfrak{u}(1,1)))$.

Background

The paper studies symmetry breaking operators between holomorphic discrete series representations for a symmetric pair $(G,H)$ and distinguishes operators represented by arbitrary differential operators from those represented purely by derivatives in directions normal to the embedded symmetric domain $H/L$ in $G/K$. For first-order operators, this distinction is encoded by the subspaces $\mathcal V^{(1)}\cap\mathcal L_{W,H}^c$ and $\mathcal V^{(1)}\cap\mathcal U(\mathfrak h_0)W$.

For the pair $$(\mathfrak{su}(n,1),\mathfrak{s}(\mathfrak{u}(n-1)\oplus\mathfrak{u}(1,1)))$$ and a nonscalar lowest $K$-type $(\tau,W)$, the authors establish that the normal first-order operators form a nontrivial proper subspace of all first-order symmetry breaking operators, but do not determine its full size. The unresolved issue is therefore to characterize or compute that subspace.