On Tuesday, Gilbert's teammate Julian Alaphilippe of France won the 16th stage of the race, his second stage victory of the Tour.
Yellow jersey holder Geraint Thomas (Sky) crossed the finish line almost nine minutes in arrears after yet another stage in which his race lead did not come under threat.
Team Sky's Geraint Thomas continues to lead by one minute and 39 seconds from team-mate Chris Froome, with Team Sunweb's Tom Dumoulin a further 11 seconds behind. "We're trying to remain dignified, we're trying not to react, and we're trying not to get distracted by it". "I think he's doing just fine". We can't recon all of these stages and you don't know what's coming up on some of these corners, but it must have been a bit damp and I came down and that's all there is to it.
"Chris is fine", said a Team Sky spokesman.
"'G' (Geraint) has ridden such an fantastic race and he deserves to be in yellow - and, fingers crossed, he holds it now until Paris".
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Brailsford, though, was quick to credit Froome for his support of Thomas throughout. To spice things up, as the stage begins with a climb, organisers have decided that the riders will line up at the start in a grid formation based on the overall standings. "When you come so close to winning a stage of the Tour, it's pretty devastating", added Yates, who won the young rider classification in the Tour two years ago.
Friday's stage is the final day in the mountains, but has a downhill finish in Laruns.
But the Dutchman did his best not to give up hope of catching Thomas. "Now, I just have to be happy with the fact that Froome cracked".
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"Obviously I didn't want him to have a bad day, like he did".
"I don't [feel exhausted from the Giro], I'm still riding strong, so I'm happy with my Tour so far".
Pictures also emerged which suggested Thomas' late dash for third place was nearly thwarted by a spectator who reached over the barriers to try to grab him. Behind him, with less than a minute separating Froome, Dumoulin and Roglic, the battle for the podium positions will be intense.
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L'an dernier, déjà à la gare Montparnasse, le trafic avait été paralysé ce même week-end en raison d'une panne de signalisation. "Notre objectif est de maintenir 60% à 70% des trains en circulation" répartis entre les gares de Montparnasse et Austerlitz.
But Froome's bid to join Jacques Anquetil, Eddy Merckx, Bernard Hinault and Miguel Indurain on the five-time winners list may have to wait another year.