One set of numbers is clearer: From 1998 to 2018, women were fined for coaching violations at Grand Slam tournaments more than any other infraction and are fined for breaking coaching rules more often than men. Through the first three majors of 2019, that trend has held, with 12 fines for women and six for men .

While the rule against in-match coaching stands, Allaster said the Open would encourage its umpires to give soft warnings “wherever possible” and to reserve penalizing players for coaching for the most overt cases.

“It would need to be very clear-cut and the official has seen it and the communication has been received by the athlete,” she said.

The signal from Williams’s coach was obvious, and he admitted making it, but Williams, who was behind in the match and struggling in the face of Osaka’s poise and controlled power, has denied she saw it. Still, a player does not have to see or hear coaching for it to be a violation; according to Grand Slam rules, it is the coach’s behavior that counts.

Mouratoglou said in the aftermath that he was doing the same thing as “100 percent of the coaches in 100 percent of the matches all year long,” yet he also insisted it was the only in-match signal he has sent Williams during their seven years working together.

The belief that illicit coaching is widespread is at the heart of the argument for legalizing it. So is coaching’s supposed appeal to television . Mouratoglou, for one, says that allowing coaching could be a way to help address one of tennis’s biggest concerns: an aging fan base.