Would the government really care? The target from both sides seems to be "hard working individuals" with the context being ground level workers in the basic rate bracket. Not middle management types riding the BIK gravy train.
Given they have not been able to bring themselves to apply the fuel duty accumulator policy they implemented for the last 11 years or put back on the the 'temporary' 12 month 5p cut they took off 2 years ago and they have just kicked the can down the road for another year.
Meanwhile train fares go up like clockwork every year.
On this basis, I'd say no, they are not going to simultaneously annoy both the vocal motorist and environmental nut job at the same time over the same issue.