Let $A$ be a commutative noetherian local ring, and let $D$ be a dualizing complex over $A$. Let $i$ be the minimal integer such that $H^i(D) \ne 0$ (I am assuming cohomological grading, so the differential is of degree $+1$).

Must the $A$-module $H^i(D)$ have full support? that is, is it true that its support is equal to $\operatorname{Spec}(A)$?

This is true if $A$ is Gorenstein (obvious) or Cohen-Macaulay (the canonical module has full support). But is it true in general?