The pair clashed at Yankee Stadium last week following Liverpool's 1-0 win over Wrexham in New York, with Kostas Tsimikas, who had been handed the captain's armband instead of Jones after Szoboszlai was substituted, also drawn into the exchange.
Under the leadership hierarchy established since Jürgen Klopp's time at the club, the armband passes to a player in the designated leadership group, or to the longest-serving player on the pitch if none remain, and Jones, at the club since he was nine, might reasonably have expected to inherit it.
Video of the confrontation showed Jones visibly frustrated as he spoke to Szoboszlai, before Tsimikas threw the armband to the ground, sending the online fanbase into a frenzy of speculation.
Szoboszlai played down any suggestion of a genuine fallout: "It was never a debate between us three. We just discussed something on the pitch.
"Actually, you guys make it very big. But next time we will manage to do it in the changing room. No, nobody was angry at no one.
"It's all cleared. We are cool, everybody."
New manager Andoni Iraola inherits the episode as an early test of dressing-room management, with Liverpool yet to formally name a vice-captain following Robertson's departure, and Jones separately still the subject of transfer interest from Inter Milan this summer.
Szoboszlai also used the same press conference to clear up an earlier slip of the tongue, when listing team-mates due back from their post-World Cup breaks. Discussing Alexis Mac Allister, he had said "if Macca comes back," a phrase supporters seized on as a hint the midfielder might be leaving.
"That's my English, guys," Szoboszlai clarified. "It's not 'if', it's 'when' he comes back. So now I make it clear. Can you post this as well? Because it's not 'if'. He's coming back and he's going to be part of the team as much as I know."