I think he's right you know.

What might surprise you to learn is how much material you require to fill such a depression. Suppose half the area is an average of 2" too low. Doing it in metric as that's easier, we get
half the lawn = 6m x 9m = 54 sq m
x 0.05m depth material
= 2.7 cu m material
the density of soil is perhaps around 2, so you are talking about at least 5 tonnes of material.

You might like to redo the above calculation once you have a clearer measure of the size of the depression.

I once bought two loads of topsoil, which were each 0.7 cu m, to raise the edge of my front lawn to conceal the edge of the new drive which had been laid and which had come out a bit higher than the old drive. I was surprised what little effet this material had once spread out on the ground. The slope of the edge of my lawn up to the new drive is therefore steeper than I would like it. I really needed about 3 times as much to achieve what I wanted. In my case I substantially killed off the old grass, as it was full of nasty weeds, and reseeded, at least where topcover was fairly deep. 8 years later it is still the best part of my front lawn...