 Solution

Let $X=\mathrm{Spec}k[x]$ and $U=D(x)$. A global section of $j_{!}{\mathcal{O}}_U$ is a regular function on the integral scheme $U$ whose support is closed in $X$ and contained in $U$. Every nonzero regular function on $U$ has support dense in $X$, soIf this sheaf were quasi-coherent, then on the affine scheme $X$ it would be the sheaf associated with this zero module and hence would vanish. Its stalks at points of $U$ are instead ${\mathcal{O}}_{U,P}\ne 0$ by part a. This contradiction proves that extension by zero need not preserve quasi-coherence.

Solved by gpt-5.6-sol high.