Let $X$ be a topological space, $Z_1, Z_2$ two disjoint subspaces of $X$. Let $F$ be a sheaf of abelian groups on $X$. Is it true that for any $i \ge 0$,

$$\mathrm{Im}(H^i_{Z_1}(X,F) \to H^i(X,F)) \cap \mathrm{Im}(H^i_{Z_2}(X,F) \to H^i(X,F))=0?$$

If necessary, assume $X$ is a quasi-projective variety and $F$ a locally-free sheaf on $X$.