Combine box shadow with a signed distance field

I’m using 2d signed distance fields in my current project for drawing shapes (example: https://www.shadertoy.com/view/4llXD7) and I want to add drop shadows to them. At first I thought I could exploit the sdfs distance property to achieve good enough looking drop shadows, but I found that in many cases this left undesirable results in the interior of the shapes where there is often a darker ‘x’ in center.

I found this shadertoy https://www.shadertoy.com/view/tdSGDc which is produces exactly what I am looking for in terms of quality and performance, but I can’t seem to figure out how I would combine this 1d box blur effect with arbitrary sdf shapes to achieve a nice looking drop shadow. Is there a way I can combine these two techniques to get great looking shadows?