The background check arrived today from Indiana. All it says is "A thorough search of our files by NAME, DATE OF BIRTH, SEX, RACE, and FINGERPRINTS DOES not reveal a criminal history record …"
I forwarded it on to the Indiana Secretary of State for an apostille, which fortunately is free (unlike most states).
Unfortunately, the background check did not have a signature or a seal on it, so I don't know if it will be accepted for an apostille. So in parallel, I wrote the Indiana State Police again for another copy of the background check, hopefully with a signature and a seal this time.