Annihilator of $Ext^g_R(R/I,R)$

We investigate the higher divisorial ideal $D(I) := Ann(Ext^g_R(R/I,R))$ associated to an ideal $I$ of grade $g$. Our main focus is the containment problem $ D(I) \subseteq \overline{I}$. We show that this inclusion holds for broad classes of ideals, including unmixed ideals of finite projective dimension over $3$-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under suitable homological conditions. If $R$ is Cohen--Macaulay local, $I$ is height-unmixed, and suppose $pdim_R(R/I)