Localness and extension conjecture for Dolbeault cohomology homomorphisms

Suppose $(G,G')$ is in the setting described above, with open subsets $Y\subset X$ of $G_{\mathbb C}'/Q_{\mathbb C}'\subset G_{\mathbb C}/Q_{\mathbb C}$. Let $V$ and $W$ be finite-dimensional representations of $Q_{\mathbb C}$ and $Q_{\mathbb C}'$, respectively, and let $H_{\bar\partial}^S(X,\mathcal V)$ and $H_{\bar\partial}^S(Y,\mathcal W)$ denote the corresponding Dolbeault cohomology spaces. A holomorphic differential operator is understood with respect to a holomorphic embedding or map as specified below.

Localness and extension conjecture. Any continuous $G'$-homomorphism

$$H_{\bar\partial}^S(X,\mathcal V)\to H_{\bar\partial}^S(Y,\mathcal W)$$

is given by a holomorphic differential operator with respect to a holomorphic embedding $Y\hookrightarrow X$. Moreover, any such operator defined on the indicated open subsets extends to a $G_{\mathbb C}'$-equivariant holomorphic differential operator with respect to a holomorphic map between the flag varieties

$$G_{\mathbb C}'/Q_{\mathbb C}'\hookrightarrow G_{\mathbb C}/Q_{\mathbb C}.$$

The statement is proposed as a possible extension of the localness and extension theorem for holomorphic functions to Dolbeault cohomologies, with the aim of geometrically realizing Zuckerman's derived functor modules. The supplied text gives no resolution status.