Is the cokernel of the pullback-pushforward of a coherent sheaf in the image of the pushforward on the complement

Question

Let $V$ be a projective variety. Let $W \subseteq V$ be a projective subvariety. Let $U$ denote the complement of $W$ in $V$. Denote by $i \colon W \hookrightarrow V$ and $j \colon U \hookrightarrow V$ the canonical inclusions. Let $\mathcal{F} \in \mathrm{Coh}(V)$. My question:

Does $\mathrm{Coker}(j_*j^*\mathrm{Id}_\mathcal{F})$ belong to $\mathrm{Im}(i_*(-))?$