I think if you ask H you won't get a straight answer. Cheaters lie. You need something more concrete to make a final conclusion if it is H or not. Just keep an eye open for other strange behavior from H. Hope for the best, plan for the worst.
Did H send any kind of message to OW to end all contact? That is worrisome that he won't send the letter.
One strategy I took when I had trouble getting to the truth was to contact the OM directly. I played good and bad to get him to cough up something. At one point I got him so angry at me that he gave up a lot of details trying to inflict some revenge. He sent pictures of them together at parks and other details of the conversations. Things my W lied about even after I showed her the pictures. I think some OM/OWs are just begging for a chance to tell the LBS what is up to get them out of the way. But this is a risky move too and I don't think is something most here would advise. But I'm not a passive person, I am the type to take action.
I also used my W phone to send messages to her friend who knew what was going on. She gave up a few gems too when she thought I was my W. (my W gave me full access to her phone she just didn't think her friend would be that stupid, ha ha). No more contact with this friend. BTW she is a serial cheater, cheaters love company of other cheaters. Does H have any friends that are cheaters? Beware.
Maybe send a cease and desist letter anyway. No need to sign it. See what happens? Stirring the pot has a way of getting things moving. Your H's hesitation says a lot.
Of course a lot of things I did are not DB type actions since I didn't know this site then. But they got a lot of things answered. I couldn't stand knowing I was being lied to and not being able to get to the truth. Agree or not with the methods, I got the job done.
Remember to GAL!!
Me:49 W:45 M:19 T:22 EA confirmed and ended 8/2014 S:19,17 D:9,5