Yahoo may need to go back to the drawing board

May 7, 2012, 9:30 AM UTC

Yahoo CEO Scott Thompson

FORTUNE — Nine months ago, we all thought the thing Yahoo needed most was somebody to fill the leadership vacuum left by its former CEO Carol Bartz, who sure went out with a bang. And yet, the beleaguered tech giant may find itself on the verge of yet another talent search, depending on how its latest snafu plays out.

Last week, activist investor Daniel Loeb’s hedge fund Third Point published that it had unearthed a glitch in the resume of Yahoo’s (YHOO) current CEO Scott Thompson, who has only been in charge of the company since the beginning of the year. In several locations, including Thompson’s bio pages on Yahoo and Paypal’s websites, it says that he earned a degree in computer science from a small Catholic school near Boston called Stonehill College. Unfortunately for Thompson, the college didn’t offer a computer science degree until 1983, after Thompson graduated.

Whoops. Pretend degrees are tough to justify. Yahoo has responded by saying that Third Point’s accusation is correct, that Thompson only has a degree in accounting, but the company told AllThingsD that the resume padding was due to an “inadvertent error,” and however that extra B.S. slipped in, Thompson is still qualified to do his job.

Whether or not that may be true, Yahoo must tread carefully. It has been suffering from leadership problems for so long now that shareholders are probably not thrilled with the prospect of another CEO search. But there are serious cons to keeping a leader that people may view as shifty.

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“A healthy, prosperous company that’s making money for its shareholders can afford to tell their critics to flake off,” says Mark Jaffe president of executive search firm Wyatt & Jaffe.

With its recent struggles, Yahoo is not in that position. And this isn’t just an internal leadership issue for the tech company, Jaffe says. Yahoo runs the risk of losing both consumer and advertiser confidence. “It’s a mindshare thing, their brand being very much in the public domain. As such, perception becomes nine-tenths of the game.”

Thompson seems to have developed a relatively solid vision for the company. Analysts have painted him as on the right track, if vague, when it comes to developing solutions to the company’s most pressing problems. “CEO Scott Thompson is taking the right steps in reducing costs, streamlining focus, bringing in new executives, and pruning activities outside of Yahoo’s core,” said one report from BGC Partners. “That said,” the report continued, “there are no fast fixes.”

But trust issues could wash out Thompson’s positive moves. It’s embarrassing, but the company could stop the flood of negativity if it fesses up to its mistakes. Jaffe says, “My verdict: cut the losses and move on to a better, more credible savior.”

It seems that activist investor Daniel Loeb would agree. In a letter to Yahoo’s board of directors, sent on Friday, Loeb called for Thompson’s resignation, and also loops in Patti Hart, the woman in charge of the executive search that found him: “Irreparable damage to Yahoo!’s culture will continue every day that the Board allows Mr. Thompson and Ms. Hart to remain at the helm of the Company after having clearly demonstrated that they lack even the ‘minimum qualifications for service as a director of the Company,’” the letter said.

“Mr. Thompson, in particular, cannot possibly have any credibility remaining with the all-important Yahoo! engineers, many of which earned real – not invented – degrees in computer science.”

Those are strong words from a tenacious man, says Gene Grabowski, executive vice president at Levick Strategic Communications. “I think [Thompson’s] days are numbered,” he says, even though in other circumstances, he could sweep a similar issue under the rug. “I’ll tell you why: Loeb is not going to let this go.”

This latest development adds fuel to an argument that’s been going on for months. This past February, Loeb agitated for Yahoo to appoint him and other colleagues from Third Point to the company’s board. Third Point has quite a bit of clout, as it owns 5.8% of Yahoo’s shares.

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Loeb’s ownership in Yahoo brings up another twist in the tale: after this public fight, Yahoo risks a loss of value, no matter what the company decides to do with Thompson. If he stays, shareholders may question Thompson’s leadership. If he goes, shareholders might question Yahoo’s ability to find another leader. “It does seem kind of odd that Loeb would persist in this and risk damaging the company so egregiously when his fund owns almost 6% of the company,” Grabowski says. “It’s a scorched earth approach. It just strikes me as almost self-destructive.”

This type of firefight is typical of Loeb — he buys a solid stake in a company and then raises hell in an aggressive attempt to turn it into an entity that can provide value for shareholders.

For better or worse, Loeb’s bid to get on the board was spurned. Now, he and his firm are examining Yahoo’s leadership with a fine-toothed comb. Although the comb didn’t need to be that fine-toothed to pull up Thompson’s resume discrepancy. The firm found it, Third Point’s letter claimed, by means of a “rudimentary Google search.”

Whatever Yahoo decides to do, it needs to tighten up. Plenty of business leaders likely have inflated sections in their resumes. But the last person you want doing your background check is a thwarted Daniel Loeb.