You won't know until Congress passes a law so it can be challenged in the courts. No one is saying it will be "easy" but it is not a slam dunk that only a Constitutional amendment can change it. There are some very legitimate arguments as to why only a law by Congress is necessary to define what is meant by citizenship.
If you really want to get informed about birthright citizenship, I suggest the following overview: Birthright Citizenship in the United States: A Global Comparison
There is an argument that we could try to make that Illegals are not “subject to the Jurisdiction...” of the United States because they entered the country in violation of federal law and are in continual evasion of that law as long as they illegally remain. Then one could try to distinguish Wong Kim Ark on the basis that his father was a legal immigrant. However, most likely we would lose at the Supreme Court.
This is why I think that Sen. Cruz is right to say that we should try to change the law and modify the Amendment because there are serious legal arguments on both sides of this issue. It just bugs me when people quote Levin, assume this is a slam dunk for our side, and then criticize Cruz for being a good lawyer and recognizing that most likely the courts would rule against us.