TITLE: The most outrageous (or ridiculous) conjectures in mathematics
QUESTION [191 upvotes]: The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.
An outrageous conjecture is qualified ONLY if:
1) It is most likely false
(Being hopeless is NOT enough.)
2) It is not known to be false
3) It was published or made publicly before 2006.
4) It is Important:
(It is based on some appealing heuristic or idea;
refuting it will be important etc.)
5) IT IS NOT just the negation of a famous commonly believed conjecture.
As always with big list problems please make one conjecture per answer. (I am not sure this is really a big list question, since I am not aware of many such outrageous conjectures. I am aware of one wonderful example that I hope to post as an answer in a couple of weeks.)
Very important examples where the conjecture was  believed as false when it was made but this is no longer the consensus may also qualify!
Shmuel Weinberger described various types of mathematical conjectures. And the type of conjectures the question proposes to collect is of the kind:

On other times, I have conjectured to lay down the gauntlet:  “See,
you can’t even disprove this ridiculous idea."

Summary of answers (updated:  March, 13, 2017  February 27, 2020):

Berkeley Cardinals exist

There are at least as many primes between $2$ to $n+1$ as there are between $k$ to $n+k-1$

P=NP

A super exact (too good to be true) estimate for the number of twin primes below $n$.

Peano Arithmetic is inconsistent.

The set of prime differences has intermediate Turing degree.

Vopěnka's principle.

Siegel zeros exist.

All rationally connected varieties are unirational.

Hall's original conjecture (number theory).

Siegel's disk exists.

The telescope conjecture in homotopy theory.

Tarski's monster do not exist (settled by Olshanski)

All zeros of the Riemann zeta functions have rational imaginary part.

The Lusternik-Schnirelmann category of $Sp(n)$ equals $2n-1$.

The finitistic dimension conjecture for finite dimensional algebras.

The implicit graph conjecture  (graph theory, theory of computing)

$e+\pi$ is rational.

Zeeman's collapsing conjecture.

All groups are sofic.


(From comments, incomplete list) 21. The Jacobian conjecture; 22. The Berman–Hartmanis conjecture 23. The Casas-Alvero conjecture 24. An implausible embedding into $L$ (set theory). 25. There is a gap of at most $\log n$ between threshold and expectation threshold (Update: a slightly weaker version of this conjecture was proved by Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Jinyoung Park!; Further update: the conjecture was fully proved by Jinyoung Park and Huy Tuan Pham ). 26. NEXP-complete problems are solvable by logarithmic depth, polynomial-size circuits consisting entirely of mod 6 gates. 27. Fermat had a marvelous proof for Fermat's last theorem. (History of mathematics).

REPLY [3 votes]: I think the Erdős-Szekeres conjecture (see the Wikipedia article on the Happy Ending Problem), which says that the smallest $N$ for which any $N$ points in general position in the plane contain the vertices of some convex $n$-gon is $N=2^{n-2}+1$, is a little bit ridiculous in that there is not a lot of reason to believe it. Indeed, only the cases $n=3,4,5,6$ are known, and while it is known that $2^{n-2}+1$ is a lower bound for this $N$, I really am not aware of any other substantial evidence in favor of this conjecture other than that it's a very nice pattern.