Monday night, on The Tonight Show with Jimmy Fallon, the GOP Presidential hopeful stepped up her effort to paint her time at the top of one of the nation’s largest technology companies as a success. Fallon remarked that her rise from a secretary at a small real estate company to the head of HP (HPQ) was remarkable, but he also mentioned critics who said she “kind of left the company in disarray.” Clearly Fiorina’s time at the top of HP is complicated, and as she has risen in the polls, so too has the scrutiny of the job she did.

HP’s stock price significantly underperformed rivals during Fiorina’s time as CEO, and she was eventually fired in 2005, because the merger she pushed through with Compaq was considered a failure at the time. But on other measures like revenue and cash flow growth, HP under Fiorina seems to have done better—though still not that much better than rivals. At least one HP board member from the time, Tom Perkins, now says firing Fiorina was a mistake.

On Monday night, Fiorina told Fallon that HP was a troubled company when she took over as CEO, and she was “hired to save it.” And she said she did, turning around the company’s market share problems. “We went from losing market share to gaining it. By the time I left, we were the leader in every market segment, every product category,” Fiorina told Fallon.

But even Fiorina’s claims about market share don’t really back up her claim that she was a great CEO. By the time Fiorina left HP, the company was not the “leader in every market segment, every product category.” It wasn’t actually even the leader in personal computers, which was at the heart of the acquisition of Compaq, and one of the company’s largest business segments.

What is true is that when Fiorina showed up, HP was a laggard in the PC computer market, ranking fourth behind Dell, IBM (IBM), and Compaq, which at the time was the leader. But HP’s market share had been consistently growing, not shrinking. HP did shoot up to to No. 1 in the PC market in 2001, but that was following the merger with Compaq, which before the deal had 13% of the PC market to HP’s 7%.

But then the year after the merger, HP’s PC market share dropped to just over 14%, not much more than Compaq had alone. And by 2003, HP had slumped to No. 2, trailing Dell. In her final full year at the company, Fiorina’s HP trailed Dell by almost two percentage points in the PC market.

HP did eventually regain the lead in the PC market, but it didn’t happen until the final quarter of 2006, nearly two years after Fiorina had left HP. And even by then, it was starting to look like a Pyrrhic victory. The PC market, the one Fiorina was in part chasing with the merger with Compaq, was already becoming vastly less profitable than it used to be.

It’s not really clear how much being a good CEO is a clear indication of whether one will be a good president. But it is a lynchpin of Fiorina’s campaign, and she is not afraid to bring it up. If she wants to really prove her time at HP was a success, however, Carly Fiorina is going to have to come up with a much better defense.