gimp - How to fade an image inward from the edges without loss

Using GIMP, I wish to fade an image inward from the edges, blending it with the neutral (white) background.

I have tried numerous things including feather edges, blur and Gaussian blur, but none of them produces a good result. Specifically, they all appear to blend the image outward or in both directions. I wish to fade the image inward from the edges, and without loss.

A tedious manual technique that I believe might work would be to:

1. select the background


2. grow the selection by 1 pixel


3. adjust the brightness, then


4. repeat 2. and 3. as many times as necessary

However, it strikes me that there is probably a tool that performs the same function.