Carolina Garcia beat and then kissed her bitter rival Alizé Cornet – as the French do so well – to reach the quarter‑finals of her home grand slam in one of the tournament’s most intriguing matches on Monday.

The world No27 was too powerful for her compatriot over an hour and a half on Court Philippe Chatrier, winning 6-2, 6-4 to earn another fascinating match-up with Katrina Pliskova, who spent 20 minutes longer on Court Suzanne Lenglen seeing off the lowest ranked player left in the draw, the world No97 Verónica Cepede Royg 2-6, 6-3, 6-4.

Had Pliskova, the world No3, failed to get past the stubborn Paraguayan she would have blown a chance to leave Paris as the new world No1. If she reaches the final, the prize will be becoming the first Czech to reach the top of the women’s rankings.

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While that match bristled with the possibilities of a major upset almost to the very end, the greater theatre was playing out on the tournament’s main court, packed on a windy, sometimes cool afternoon with Parisians and others who cheered wildly throughout for their chosen favourite.

The rivalry between the two French combatants was more than the usual professional stand-off. When Garcia withdrew from the French Federation Cup team last month with a back injury – an injury which Cornet, along with team-mates Kristina Madlenovic and Pauline Parmentier regarded as unconvincing – they engaged in an entertaining Twitter spat that had French tennis amused for days. The only one not amused was Garcia. She said before this fourth-round match she had no intention of reconciling their differences, before or after and regardless of the result. Predictably, they met at the net and exchanged what appeared to be at least formal cordialities, a cheek-to-cheek peace kiss that might have buried their feud.

Garcia, meanwhile, will turn her attention to the bigger challenge awaiting her against Pliskova on Tuesday.

The Czech has had a decent season, playing lots of matches, and came to the tournament full of confidence – so much so, the Guardian was inclined to regard her as having a good shot at the title in a field without Serena Williams and which, after the first week, had also lost the defending champion Garbiñe Muguruza and the world No1 Angelique Kerber, who went out in the first round.

If Pliskova rediscovers her wicked serve – she arrived with 221 aces from 35 matches, the best tally on the Tour – she will be hard to beat, even though she sometimes struggles on clay.

Simona Halep is another one of the game’s stormy presences – as she admitted after an on-court row with her coach Darren Cahill in Miami – but she declared herself happy and upbeat here on Monday when, reunited with the Australian, she demolished Carla Suárez Navarro in an hour to advance to the quarter-finals.

The combustible Romanian has had another interesting ride through the first week of the tournament. Two Fridays ago she was unsure if she could play at all, fearing torn ligaments in her ankle. There followed similarly gloomy prognoses until last Thursday when she declared: “I’m fit. I’m ready to play three hours. I don’t care about that.”

Halep turned her right ankle on Saturday, but insisted: “It doesn’t bother me much. I’m not thinking about it. I can run, I can slide. I have nothing in my mind about injury anymore.” After beating the Spanish world No21 6-1, 6-1, Halep was beaming. “I feel very, very good now. About my ankle, I still feel some pain, but it doesn’t bother me. I can run. I’m not thinking about it during the points. I feel that I’m fit. I’m ready to go in this second week.”

As for her renewed partnership with Cahill, she said: “Everything is changed, also in my general life. I’m more positive, and he helps me. I don’t have bad moods anymore. It’s good. Now I’m OK with the way I am on court and out of court.

“My attitude: I knew that is the only one thing that I have to change to have him back. So I work hard, and I changed. I changed pretty fast.”

In the quarter-finals, Halep plays Elina Svitolina, who laboured for more than two hours to beat the Croatian qualifier Petra Martic, 4-6, 6-3, 7-5.