There is a legal doctrine of Promissory Estoppell where you can enforce a promise if you changed your position in the light of the promise, relying on the promise to your detriment.
If you didn’t rely on the promise to your detriment, then you cannot enforce the promise.
If example you were going to build a huge shed or suchlike and you said that you would not build a huge shed provided he promised to change his extension plans, so you didn’t build a shed, you could enforce the promise.
Other than that, you are faced with objecting to the application on the grounds of the impact on your property.
The neighbour is not under any duty to discuss it with you directly.
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I can still clarify things for you.
I’m sorry. I was with a client.
For there to be an agreement to do something, there has to be an offer and acceptance and consideration and it’s unlikely that the text would satisfy the criteria.
It would be proof of the promise