The Hartogs phenomenon in toric varietries

We say that a complex space $(X,\,\mathcal O)$ admits the Hartogs phenomenon if for any compact subset $K$ of $X$ such that $X\setminus K$ is connected, a restriction homomorphism
$$ H^0(X,\,\mathcal O) \rightarrow H^0(X\setminus K,\,\mathcal O) $$
is an isomorphism.
In toric varieties this phenomenon has been explored by M. Marciniak [1] who related it to properties of corresponding fans:
Theorem. If $X_\Sigma$ is a smooth toric surface with a strictly convex fan $\Sigma$ then $X_\Sigma$ admits the Hartogs phenomenon.
She has also formulated a conjecture for toric varieties of higher dimensions: A smooth toric variety $X_\Sigma$ admits the Hartogs phenomenon if the complement of its fan $\Sigma$ has at least one concave connected component.
We were able to prove it not only for smooth but also for normal toric varieties. Let $X_\Sigma$ be a normal toric variety corresponding to a fan $\Sigma\subset \mathbb R^d=\mathbb Z^d\otimes_{\mathbb Z}\mathbb R$. We shall say that a connected component of $\mathbb R^d\setminus |\Sigma|$ is concave if its convex hull coincides with $\mathbb R^d$.
Theorem. Let $X_\Sigma$ be a normal toric variety. If the complement of its fan $\Sigma$ has at least one concave connected component then $X_\Sigma$ admits the Hartogs phenomenon.
The proof follows from the study of Dolbeault cohomology with compact support of a smooth toric variety where $X_\Sigma$ can be equivariantly embedded.
This is a joint work with S. Feklistov.