That revelation, which Abreu made last year at the smuggling and conspiracy trial of an agent and trainer, underscored the desperation of Cubans to leave a country that prevents them from seeking their market value. Abreu has made good on the White Sox’ investment; he is the only player in the majors to hit at least .290 with 25 homers and 100 runs batted in in each of the last four seasons. Listening to Minoso, he said, helped his transition.

“When I first came here, he was a person who gave me the trick to have success in this country, and in the major leagues: work hard, respect and punctuality,” Abreu said. “Those are the three keys I’ve been trying to pass to Yoan and Robert, and all the people around me, too, because I know it’s true. I’ve applied them, and I’ve enjoyed the success I’ve enjoyed because of it.”

The White Sox have changed over nearly their entire roster since Abreu arrived. They have not reached the playoffs since 2008 and tried veteran imports for years, even as recently as 2016, when they dealt Fernando Tatis Jr. — now considered one of baseball’s 10 best prospects — to San Diego for James Shields, an expensive starter who has gone 9-19 for Chicago.

But Hahn had a ticket out of persistent mediocrity: modestly paid, high-impact players with multiple years of club control. From December 2016 through last July, he traded Sale, outfielder Adam Eaton, starter Jose Quintana and relievers Tommy Kahnle and David Robertson for an avalanche of prospects who have lifted the White Sox from 23rd in Baseball America’s 2016 farm system rankings to fourth (behind Atlanta, the Yankees and San Diego) this spring.

Now, they wait.

“The final stage, where you start adding to this young core at the big-league level with premium free-agent talent, that’s exciting, because you’re getting close,” Hahn said. “It’s this in-between stage — when you have to show patience and let player development take its course — that’s perhaps the most challenging of the whole thing.”