But now, it turns out that what some thought was an endorsement wasn't -- according to Ray Genco, an attorney for Rose -- and that Trump's ball didn't come from Pete Rose at all. Genco made this absolutely clear in a telephone conversation with The Washington Post late Monday.
"We do not know how Mr. Trump got the ball," Genco said. "I can't authenticate the ball from some Twitter picture." He added: "I can't speak to how Trump got the ball. Pete didn't send it. I made that clear."
Genco's verbal clarification followed a statement he sent to The Post and other outlets Monday.
"Pete has made a point not to 'endorse' any particular presidential candidate," Genco wrote. "Though he respects everyone who works hard for our country - any outlet that misinterpreted a signed baseball for an endorsement was wrong. Pete did not send any candidate a baseball or a note of endorsement. That said, through my discussions with Pete about this cycle, I've learned that he believes that who to vote for is a decision each voter should decide for him or herself. Pete knows and has impressed upon me that, above politics, it's leadership and teamwork [that] make all the difference. Both the left and right are Baseball fans -- and it is those institutions and their people that make America exceptional."
Eyes now turned to the Trump campaign. Question No. 1: Was Trump trying to imply that Rose had endorsed him when Rose hadn't? The Post asked.
"Did Mr. Trump claim this was an endorsement?" Trump spokesman Hope Hicks replied in an email. "He was just thanking Mr. Rose for the thoughtful gesture."
Question No. 2: Did Trump get the ball from Pete Rose, or as Trump's tweet put it, was it "received" from Rose? If not, where had he gotten it? Did the "gesture" really come from Rose at all? Asked such questions in a follow-up email late Monday, Hicks did not reply.
Further complicating the story is Rose's penchant for signing a lot of baseballs. Walmart sells a "Pete Rose Autographed I'm Sorry I Bet On Baseball Ball With Psa/Dna Authenticity" for $125.78. And as the Wall Street Journal reported in 2013, the man himself hawks them in Las Vegas.
"Since 2005, he has spent several hours per day, 15 to 25 days per month, 12 months per year signing his name for money," Brian Costa wrote. "He signs and poses for photos with anyone who buys memorabilia from his business partners, with items such as baseballs and photos ranging from $75 to $800. And for this, Rose earns more than $1 million per year."
"Pete's job is to be Pete Rose," Joie Casey, president of Hit King Inc., which manages Rose's autograph ventures, told the Journal. "And he's the best Pete Rose there's ever been."
After Rose's non-endorsement, some shared photos of balls purportedly signed by Rose on Twitter. Some of the messages, which included "I'm sorry I shot J.F.K." and "I was the 1st man on the moon," were transparently ridiculous.
Also complicating the story are some candidates' ability to take words out of context to falsely whip up what looks like an endorsement. Sen. Bernie Sanders of Vermont, for example, was dinged for this in New Hampshire last month after an ad made it seem as though newspapers that had not endorsed him had done so.
"The words 'endorsed by' then reappear next to the logo of the Valley News, which appears along with the quote: 'Sanders has been genuinely outraged about the treatment of ordinary Americans for as long as we can remember,'" PolitiFact wrote of the ad. "So did the Valley News or the Telegraph endorse Sanders? According to their editors, no."
And even further complicating the story is Trump's sometimes shaky relationship with the truth. Just months ago, PoliFact gave "the campaign misstatements of Donald Trump" its "2015 Lie of the Year" award.
"It's the trope on Trump: He's authentic, a straight-talker, less scripted than traditional politicians," it wrote. "That's because Donald Trump doesn't let facts slow him down. Bending the truth or being unhampered by accuracy is a strategy he has followed for years."