Local exactness conjecture for singular BGG complexes

The singular BGG complex is constructed on the twistor space $Z'$ associated with a suitably chosen, but arbitrarily small, open subset $X'$. Local exactness conjecture. The conclusion of the vanishing lemma is true for the twistor space $Z'$, where $X'$ is a suitably chosen, but arbitrarily small open subset. This conjecture would imply local exactness of the singular BGG complex, meaning exactness in the category of sheaves; the source provides no resolution status.