Strongest self-described socialist candidate for POTUS between Debs and Sanders?

[David T]

Who was potentially the strongest self-designated socialist candidate for the presidency of the United States between (but not including) Eugene Debs and Bernard Sanders? I emphasize the "self-designated" because the word "socialist" is notoriously difficult to define, and I do not want to get into endless arguments about whether Mr. X or Ms. Y was "really" a socialist. My requirements are simply these:

(1) The candidates must either explicitly describe themselves as socialists or be members of some group with the word "socialist" in its title.

(2) If third-party or independent candidates, they must have the potential of getting at least ten percent of the vote. (True, Debs only got six percent of the vote in 1912, but it is arguable that he could have reached ten percent if TR weren't around and it was a Taft vs. Wilson race--or better still, a race between Taft and a more conservative Democrat than Wilson.)

(3) If they are running for a major party nomination, they must have a good chance of winning at least some primaries (outside their home states).

Two people I want to rule out at once are Robert La Follette and Henry A. Wallace. It is true that La Follette was on the ballot as the Socialist candidate in several states in 1924, but that was only where he could not get on the ballot as an independent or Progressive. (He attempted to get on the ballot in California as an independent but the state Supreme Court decided against him 4-3 on technical grounds in what many regarded as a politically motivated decision. Still, even as a "Socialist", he got 33 percent of the vote in California, well behind Coolidge but far ahead of Davis.) He tried very hard (and succeeded in most states) not to have to appear only on the Socialist ticket--both because he wasn't a Socialist (he didn't consider himself a small-s socialist either) and because he realized that appearing as the Socialist candidate would limit his appeal. As for Henry A. Wallace, he was emphatic throughout the 1948 campaign that he favored "progressive capitalism" and despite the strong influence of the Communist Party in the Progressive Party of 1948, the latter's platform was not (at least explicitly) socialist.

So with those eliminated, who do we have? One possibility: Some really bad scandal in the Giuliani administration leads to Ruth Messinger--a member of Democratic Socialists of America--being elected Mayor of NYC in 1997 In 2000, Gore and Bradley divide "mainstream" Democrats, making a successful challenge from the Left somewhat plausible...