Attached my solution.

Yes, this is why any continous deformation of the loop within the field-free region will yeild the same value when integrated around.

I would add that if the field-free region were topologically trivial, you could deform the loop in this manner down to a single point (i.e. a loop of infinitesimal size). In that case the loop integral would be zero, and because the value of the integral is unchanged by such deformations, that would mean all such loop integrals would be zero.

It's the presence of the pillar (containing non-zero magnetic field), inside the loop in your diagram, that ensures the loop cannot be contracted to an infinitesimal point: The topology of the field-free region is non-trivial. This is required to prevent the loop integral from always being zero!