Hm, I think one issue here is that they have different expectations as far as what a delivery estimate should be. They want to know when they can expect it: Mar 15, +/- 4 days. You're giving them the latest it will possibly be: before Mar 23.

Both are highly useful. They can know that they will not loose face with their client if they say it'll be done on the 23rd. OTOH, they can't know if they should start planning for what the people presently working on the project should be doing from the completion -- which is going to be on the 15th, more or less.

Like everything else, agree on your terms, or state them.