Is there a constitutional case on Trump's Truth Social posts sale?

 I'm talking about Trump selling faster access to his dementia-laced addled comments.

The Intercept and the Freedom of the Press Foundation claim their is, in a new suit. Here's their case: 

The Intercept and the nonprofit Freedom of the Press Foundation alleged that the Truth Social offering runs afoul of the First Amendment's guarantee of equal public access to the president's statements and also violates the Fifth Amendment by granting preferential access for "unreasonable sums." The suit asks a federal court in New York to block the president from posting official government information exclusively on Truth Social.

Let's look at both, starting with the Fifth Amendment angle, because I pulled it up first. We normally think about "Pleading the Fifth," but the amendment has more than that, being kind of a grab-bag in a sense. Here it is; note the end:

No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation.

OK.

Is it the second-last clause, the due-process clause, or the private property clause, that's being cited?

If the last clause had reverse wording, the case would be a slam dunk. But it doesn't.

Beyond that, if the last clause is the claim, the litigants shut themselves in the foot, IMO, with the "unreasonable sums" claim. They're essentially saying they WOULD pay, but only if "reasonable."

OK, now the famous First:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

The only way I see even a possible shot here is if Trump charges some media more than others for the same expedited speed, or grants that to some outlets for free. 

Otherwise, like lawsuits over cutoffs in Corporation for Public Broadcasting funding (ironic because my story link is from NPR) I see no, NO constitutional angle at play.