Convert PDF image into true grayscale?

Question

Following the answers to this post I get the following script:

#!/bin/bash
gs \
 -sOutputFile=/tmp/output.pdf \
 -sDEVICE=pdfwrite \
 -sColorConversionStrategy=Gray \
 -sColorConversionStrategyForImages=Gray \
 -dProcessColorModel=/DeviceGray \
 -dCompatibilityLevel=1.4 \
 -dNOPAUSE \
 -dBATCH \
  $1

mv /tmp/output.pdf $1

However, running identify -verbose file.pdf | grep Colorspace afterwards shows Colorspace: sRGB.

I want to make these true grayscale PDFs, and have the Colorspace be set to gray. I have one PDF that reports gray, but I don't know how to make ghostscript convert the other ones to be like it.