It's important to understand the difference between the conflict of interest statutes and the emoluments clause of the Constitution. This suit is not based on the conflict of interest statutes, which exempt the President. Here is the exact language: “No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State.”
An "emolument" is basically defined as basically any financial benefit. The suit is based upon the fact that there are funds flowing from foreign governments to businesses owned by Trump. There was a long discussion on this before, but the only remedy for a violation of the clause by the POTUS is impeachment, which isn't likely with a Republican-controlled Congress. This would only happen if, as happened with Nixon, a sufficient number of Republican legislators were repulsed enough by his conduct to vote to impeach, and then to convict. I don't think this suit will go anywhere because, first, the court as presently constituted is evenly divided and would be unlikely to take up the question, even if it weren't, probably pointing out that there is a remedy in the Constitution. Even if it did render an opinion that he was in violation, then what? Once again, enforcement is the issue. As in Jackson's remark when the court found that Georgians had violated treaties with the Cherokees - "John Marshall has made his decision, now let him enforce it."
Edit: I see the WaPo article explains a good bit of what I've just posted...