Chanel’s chairman, Alain Wertheimer, and his brother, Gérard, each own half of Chanel and have $14.1 billion fortunes, according to the Bloomberg Billionaires Index. The pair, who rarely take front-row seats at Chanel fashion shows or talk to the news media, are known as fashion’s quietest billionaires. But one is 69 years old and the other 68, and it is unclear who will succeed them.

In 2016, the former chief executive officer Maureen Chiquet was fired because of differences of opinion over the strategic direction of Chanel with the Wertheimer family. Amid speculation about receding profits and dipping sales, her role was assumed by the chairman, whose grandfather Pierre Wertheimer (and his brother Paul) co-founded the Société des Parfums Chanel to produce and market Chanel fragrances and beauty products in 1924.

Originally credited 35 years ago with resurrecting the once-stagnating label by hiring Karl Lagerfeld as designer, and creating a luxury powerhouse led by sales of perfumes and cosmetics, Mr. Wertheimer has taken on more responsibility to “ensure continuity” in the wake of Ms. Chiquet’s departure. He is working closely with other members of the leadership team to devise a plan to secure the fashion house’s long-term future. But details were not forthcoming.

“Our C.E.O. is in great shape and seems to come up with an idea every five minutes,” Mr. Blondiaux said. “Who might come next is not on the agenda.”

Similar questions around who might replace Mr. Lagerfeld at the creative helm were also rebuffed. The flamboyant German designer — given to controversial statements such as attacking Angela Merkel and her immigration policy — refuses to divulge his age, but is now in his mid-80s.

That doesn’t mean he is going anywhere yet, though the former Céline creative director Phoebe Philo and her successor at that brand, Hedi Slimane, have both been tipped as potential successors.

The industry has obsessed over possible new ownership, as well.

Rumors have circulated with increasing intensity in recent months that, as one of the last privately owned brands in the business, Chanel was a likely target for a takeover approach.