fail to fulfil or undertake (a responsibility or duty).
"the government was accused of abdicating its responsibility"
avoid or neglect (a duty or responsibility).
"I do not shirk any responsibility in this matter"
Shirk is less formal and I'd say a bit more crude than abdicate. There may be other synonyms to those words that could be used like "evade" or "neglect", depending on the situation.
I thought of some more information that doesn't answer the question directly but could be useful culturally. If this information is being conveyed in an environment where politeness or co-operation is a priority, eg. a business, it's less common to use an accusation to describe an issue like this. This especially applies if the issue is being brought to the attention of someone for the first time or if it could be perceived that the person who made the promise is behaving reasonably.
An alternative is to describe the situation, and particularly the role the promise has in an outcome, and let the audience infer the accusation if they believe it is appropriate. An example would be "The report can be finalised once I get John's approval".