What will be the impact of restrictive immigration policies on the US housing market?

Since the 2008-09 recession, housing in the US has been a solid investment. Is there any chance this could change in the future and over what time horizon?

Mainstream media initially predicted restrictive immigration policies would lead to increased housing prices by reducing the construction labor pool. It seems possible that this could be offset by reduced demand for new housing if total immigration drops significantly and if the US continues to have a low domestic birthrate. Other policies play into overall housing price, especially in the short-term. Which will dominate?