I figured it out. The browse control requires you to select the directory, press OK, and only then will it list any files. Then you select the file and press OK.
Very non-intuitive! Every other browse or file -> open control I've ever used had you press the OK button once as the last step.
I thought I had to use .svg files because that is what was listed by default. However, I've been able to use the .xpm files or even .png files.