Does an equal $D$ (Displacement current) on two surfaces gurantee that there is no $Q$, Or is it the other way round no $Q$ hence no change in $D$?

There is a question in the first chapter of RF Harrington’s book asking to show that the $Q$ (charge density) will be zero if there is no source. I am posting the solution found from the guide.
In the solution, they assume that $D$ will be equal in both the surfaces since there is no source $Q$. However, isn’t it the other way round? (the lack of $Q$ is why $D$ is equal)