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Then how are you handling this problem? They are ofcourse fraud and justice needs to be done. I think you must reach out to the Home Quality Remodeling contractors online for the professional advice on this. They are licensed and very professional. So, they can solve your problem easily by giving you the best solution possible.
The only leverage you might have now since they are deneying it and already have their money is to start placing bad reviews of them online.
Top review sites are Google, BBB, Yelp, Houzz and Facebook. Also Angie's List and Home Advisor are good if they are on these platforms?
Give them one last chance to correct the problem along with a list these review platforms that you WILL be posting bad reviews on if it is not resolved within a stated timeline. If they are smart then they should resolve the problem.
If not, then place the bad reviews and send them a list of website links of each of the bad reviews. Let them know that once they resolve the problem then you will take all the bad reviews off.
Not much else you can do except go legal and this does not seem to warrant that?
Best to You