When the City Council has reason to believe that the owner of any property may successfully claim the same as exempt from assessment, the council may order that the improvement shall not be made in front of said property, unless the owner shall first satisfactorily secure the payment of the amount which would be assessed against the same, were it not exempt. When any part of the cost of improvement is to be paid by such property owners or assessed against their property and the contractor is required to look to such property or its owners for the proportion of the cost assessed against them, and the city is relieved from the payment of such cost, the contractor may or may not be required to construct improvements in front of property which is exempt from the enforcement of a lien for such cost. The omission of improvements in front of such property on a highway or section thereof ordered to be improved, shall not invalidate or affect any assessment against any other property located on such highway, or section thereof, or charge of personal liability against the owner thereof.