i don't think Conscientious Objectors are given an exemption, not automatically at least
it's hard to say b/c the sites are down (hahahah) but the exemptions i'm seeing are:
Transgenders
currently on Active Duty
not 18 or over 25
it you're form Micronesia, Palau, Marshall Islands
etc
edit they have to still register
CONSCIENTIOUS OBJECTORS
Men, who would be classified as CONSCIENTIOUS OBJECTORS if they were drafted, must register with Selective Service. If a draft begins and they are called, they would have the opportunity to file a claim for exemption from military service based upon their religious or moral objection to war.
https://www.selectiveservicenumber.org/who-must-register