Froome's last hope appears to rest on whether Thomas cracks in the last mountain stage on Friday, which seems unlikely after the Welshman proved stronger in all three summit finishes.
Froome's compatriot Geraint Thomas kept the yellow jersey and a three and a half minute advantage over Quintana, who couldn't find his form until Wednesday's mountain stage.
"I saw Froome was in difficulty but I didn't know if it was a bluff so I waited a bit with my attack".
Froome was among a large group of riders whose eyes needed treatment due to the tear gas. "I'm happy just to be in the position I'm in", he said.
In the bid to clear the road and detain the protesters police sprayed tear gas, which then drifted into the peloton and led to riders having to visit the medical auto to wash their eyes out.
Thomas, who has never previously been in a position to win a grand tour, is keeping his feet on the ground.
Among the riders affected by the tear gas was Team Sky's overall leader Geraint Thomas, who was pictured rubbing his nose following the incident.
Manchester City vs Liverpool
Have we played Liverpool before in the ICC? But in defensive midfield, they look light. It all underlines the challenge for City when it comes to reinforcing the position.
Yates accelerated solo on the climb but, with the QuickStep Floors rider in hot pursuit, the Brit attacked the descent hard and paid for his risk-taking when he crashed - allowing the Frenchman to sweep past.
This means Tom Dumoulin moves into second place to split the two Sky riders, Thomas and Froome, on the podium.
Race organizers have struggled to deal with angry fans protesting Froome's participation.
Before that, Gilbert posted a video online explaining why he would be unable to finish the race.
"My intention was never to criticise, just to draw some attention to the fact that I was getting pretty anxious about our guys", Brailsford added.
"That was a hard crash but, fortunately, I haven't broken anything and I feel confident about the last four stages of Le Tour de France", the man now wearing the sprinters' green jersey later posted on Facebook. "For me, so far it has not been possible to gain time on him". "I usually improve in the third week of the Tour and it's going that way".