Cracks are visible on some of the chimneys and need to be rebuilt for safety reasons and to ensure they last, say the developers.
But campaigners say taking the chimneys down will ruin the character of the building.
A small group of people dressed as monkeys gathered outside the station today with placards claiming the company behind the development 'don't give two monkeys' for the iconic building.
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A lamp in the tower which houses Big Ben is to be switched off for the first time in more than 70 years.
Sir Bruce entertained audiences for more than seven decades.
Work in the tower will also pause over the bank holiday weekend.