A while back Holmes grave was exhumed and the body inside sent in for DNA testing... results are in. Holmes was indeed executed and buried in that grave.
Chicago Tribune, 9/2/2017: http://www.chicagotribune.com/news/local/breaking/ct-hh-holmes-gravesite-exhumed-20170901-story.html
Washington Post, 9/1/2017: https://www.washingtonpost.com/national/tests-confirm-gravesite-of-19th-serial-killer-h-h-holmes/2017/09/01/25c9a502-8f32-11e7-9c53-6a169beb0953_story.html?utm_term=.14139f0288e6
NBC 5 Chicago, 9/1/2017: http://www.nbcchicago.com/news/local/American-Ripper-DNA-Reveals-Identity-of-Body-Found-in-HH-Holmes-Grave--442267363.html