I can’t launch the game using HD.

I have the exact same problem. I spent a lot of time on it, and not just my own, but also the time of the helpful people here (a separate thank you to them). But I couldn't solve the problem. So that others don't waste their time, I'll list what definitely DOESN'T help:
1. Disabling all firewalls and antivirus programs.
2. Using Hamachi and setting network priorities.
3. Installing a VPN version of TeamViewer.
4. Downloading various "correct" versions of Heroes with and without the mod pre-installed.
5. Changing computers (!).
6. Changing the connection location.

What I was able to determine: it connects with some people, but not with most. What it depends on - I don't know. Without the mod, it connects with everyone. The WoG version connects with everyone.

The fact that HDMod affects the connection is 100%, because without it, everything works, and without any complicated procedures. I contacted Baratortch in a private message and on the forum, but apparently, my number of posts on the forum is not enough to get a response.