The company will pay you the lowest amount you are willing to accept. Location has nothing to do with salary rates. If they could find someone at lower salary, anywhere in the world, they would hire that person. They can’t, that’s why they are paying the current salary rates.
The company may use it as an excuse, but you can choose to walk. When enough top level people refuse pay cut, the company will drop the charade quick. Even if just 10% of high performance engineers decide to walk, that would cause major disruptions in the roadmap and cost millions in development costs.
Simply demand higher salary and negotiate. The company will make adjustments to fill the in demand positions. Be prepared to walk away and find the company that will pay the higher salary.