(And so did Ding Liren, but he's out of the race for first.)
For most of the day, it looked like Anish Giri was the story of the round. He defeated Fabiano Caruana, with Black, in a very well-played game where Caruana was the first player to break new ground. It was a great achievement by Giri, and not only seemed to guarantee him a share of the lead with two rounds to go, but he also put an end to Caruana's hopes of achieving a second world championship match with Magnus Carlsen. (Or close. He may yet be mathematically alive, assuming there are no tiebreak problems.) All of those accomplishments deserve the praise that I and many other commentators are bestowing upon Giri, but there is one hiccup that must be noted. Ian Nepomniachtchi somehow managed to defeat Wang Hao from what had been an extremely drawish position (and Wang's premature resignation was a nice "chef's kiss" touch to his self-destruction, and all-around odd performance in the second cycle). So, despite Giri's win and his streak of 3.5 points in his last four games, he remains half a point behind Nepo heading into the final rest day.
Still alive in the hunt for first, but just barely, is Maxime Vachier-Lagrave. His performance in the second cycle had been very poor, but today he won an excellent game against bottom seed Kirill Alekseenko to stay within a point and a half of Nepomniachtchi with two games to go. More good news for MVL: he plays Nepo in the next round, but he'll have the black pieces.
Finally, as noted in the parenthetical at the top of the post, Ding Liren won, putting an end to Alexander Grischuk's hopes in the event. (Grischuk entered the round tied with MVL, a point and a half behind Nepomniachtchi.) Grischuk played solidly all game, but then played wildly and badly after the time control, speedily turning a very drawish position into a loss.
(The games, with my comments, are here.)
So, with two rounds to go, there are two favorites, Nepo and Giri, with MVL having a small chance and Caruana (at best) a snowball's chance. The players have Sunday off, and here are the pairings for the penultimate round on Monday and the finale on Tuesday. (It is technically possible for there to be a playoff, but it's unlikely, and the mere fact of a tie doesn't guarantee it - the players involved must be tied on multiple tiebreakers as well.)
Round 13 Pairings:
Round 14 Pairings:
Nepomniachtchi has a half-point lead over Giri, while Giri has the easier schedule--especially on the last day. And it doesn't help Nepo that Ding won today and gets to play the tournament's tailender and bottom seed on Monday; he may enter the last round with plenty of confidence and even a touch of ambition, if a win would bring him up to 50%. Nepo has another very important advantage, though: he beat Giri in the first cycle, so if they finish the 14th round with the same number of points (and no one else is tied with or ahead of them), Nepomniachtchi wins on tiebreaks.
In case the event finishes in a two-way tie between Giri and MVL, I'm not sure who finishes ahead on tiebreaks. The head-to-head was event, as they drew both games, but there are other tiebreaks as well. As for a two-way tie for first between Nepo and MVL, MVL would win that one on tiebreaks--but it's an unlikely scenario. MVL would have to defeat Nepo in round 13, and then outscore him by a half a point in the last round - let's say MVL wins and Nepo draws. That's not impossible, but the only way they finish in a two-way tie for first is if Giri only manages half a point in his last two games. It's possible, but unlikely. As for what would happen in case of a three-way tie for first, I'm not sure.
The most fun scenario, from a clown-car disaster perspective, comes if Nepo loses his last two games, MVL beats Nepo and draws with Wang Hao, Giri scores half a point in his last two games, and Caruana finishes with two wins. Then all four players tie for first. (Note that Caruana can also be involved in a three-way or even a two-way tie for first.)
All these conjectures are entertaining, but it's also possible that the tournament ends, practically speaking, on Monday. If Nepomniachtchi wins and Giri fails to win, Nepo clinches.