Nobody would like this kind of situation ! 

Obviously you know the expectations of the customer.  But do you know the expectation of your boss / your company ?  Is it possible that the customer's satisfaction is needed at any cost for strategic reasons, like for example to win a more important contract ?   In that case, you'd better make sure that your boss is aware of the situation and endorses it. 

If not, then you've to find out which arguments you could best use with your customer. 

Are you in a times and means contract, where the customer has to pay the time spent on the project ?   Then make it clear to your customer that he has a strong financial interest to let you manage the scope in a more focused way.   If necessary, reestimate the project budget to show the additional costs that come with the additional work. 

Is there any fixed deadline that you have to meet ?  Especially hard deadlines for your customer ?  Then show youd customer that with your team you will not be able to achieve the extra work and the SOW in the same amount of time.  Show him the extra delay that he will have to accept.  Eventually find out with him if some elements of the SOW are not needed anymore, to get a new balance.  He will then appreciate yur scope maangement.

Or are you in a fixed-price relation without any real time pressure for yur customer ?  Then you have no other alternative than to play the bad guy : you are not responsible for promises that were not explicitely agreed. Control your people to keep them  working strictly on the scope and do not hesitate to escalate the issue to the steering committee.  Keep in mind that if you don't play hard, you will be hold responsible for the mess at the end by everyone. 

A last thing you should think about.  Some customers play tricks.  Some customers invent promises, or exagerate them, when the project manager did not participate to the selling negociations.   If you've the feeling that this could be the case, you could also organise a clarification meeting together with the custome and the salesmen to find out. 

Good luck.  And give us some feed-back about the way you will have solved the issue. 

Nobody would like this kind of situation ! 

Obviously you know the expectations of the customer.  But do you know the expectation of your boss / your company ?  Is it possible that the customer's satisfaction is needed at any cost for strategic reasons, like for example to win a more important contract ?   In that case, you'd better make sure that your boss is aware of the situation and endorses it. 

If not, then you've to find out which arguments you could best use with your customer. 

Are you in a times and means contract, where the customer has to pay the time spent on the project ?   Then make it clear to your customer that he has a strong financial interest to let you manage the scope in a more focused way.   If necessary, reestimate the project budget to show the additional costs that come with the additional work. 

Is there any fixed deadline that you have to meet ?  Especially hard deadlines for your customer ?  Then show youd customer that with your team you will not be able to achieve the extra work and the SOW in the same amount of time.  Show him the extra delay that he will have to accept.  Eventually find out with him if some elements of the SOW are not needed anymore, to get a new balance.  He will then appreciate yur scope maangement.

Or are you in a fixed-price relation without any real time pressure for yur customer ?  Then you have no other alternative than to play the bad guy : you are not responsible for promises that were not explicitely agreed. Control your people to keep them  working strictly on the scope and do not hesitate to escalate the issue to the steering committee.  Keep in mind that if you don't play hard, you will be hold responsible for the mess at the end by everyone. 

A last thing you should think about.  Some customers play tricks.  Some customers invent promises, or exagerate them, when the project manager did not participate to the selling negociations.   If you've the feeling that this could be the case, you could also organise a clarification meeting together with the custome and the salesmen to find out. 

Good luck.  And give us some feed-back about the way you will have solved the issue.