How to dilate binary image from outside-to-inside?

I have a binary image of a line, like this one:

What I’d like to do is to dilate the line, but only towards the ‘inside’ of the line. That is, to add white pixels towards the center of the image, using DiskMatrix[] or any other kernel of a given radius/size. I’ve been trying to find ways to do this for a bit but I have no idea where/how to begin.

Thanks!