What does "soon after" mean for the USSC? Are we talking about a few months or a few years? Does a case need to go through a bunch of appeals from lower courts and spend many millions first?
Short answer: Probably years and millions.
Under article three, section two of the Constitution, the U.S. Supreme Court has "original jurisdiction" only in "Cases affecting Ambassadors, other public ministers and Consuls, and those in which a State shall be Party." Mostly, they only decide cases that have already been through at least two levels of federal courts, and maybe some state courts before that.
Referring, of course, to the proposed U.S. legislation which could cause severe damage to the Internet—at least, that's what a lot of people are saying. See, e.g., this Open Letter From Internet Engineers to the U.S. Congress (the first signatory listed is Vint Cerf). On Wikipedia, people including Jimbo Wales are discussing strategies as extreme as blanking the entire site (except for an explanatory message) to get people's attention, and thereby perhaps incite them to action, such as calling their Congressional representative.
I just happened to find out about all this a few hours ago, being someone who tries to avoid distractions like most kinds of news, so possibly others here with similar habits will appreciate having it called to ther attention. Or possibly they won't. But to those of you who possess relevant kinds of expertise:
(I think this subject can be discussed without political advocacy, in which I am mostly not at all interested anyway. It just looks like a practical problem to me.)
Edited to Add: I forgot to include a fourth bullet point:
It seems to have been assumed by many commenters, nevertheless.