New information! I have received a message from a user who can not download the file on their PC.
The user said this;
Hi Henriko! I am experiencing the new bug where I cannot update the pack on the technic launcher. It claims it is unable to download a certain file. My computer is running Windows 10, I live on the east coast USA, I have no VPN, and my internet is from charter spectrum. Interestingly, the computer in question belongs to my sister (we play with our cousin on a private server) and my own computer, which is also running Windows 10, seems to be downloading the update without issue, while hers (on the same internet connection) cannot. I hope this is helpful!
So, from this message we can deduce a few things:
1. A VPN probably won't make the issue occur (but it might resolve the issue)
2. It differs from PC to PC, and is NOT network related it seems.
3. The OS doesn't seem to matter.
4. It could be an anti-virus issue, which is blocking access to HFS File Server files.
So all in all, this seems to be an issue only affecting some PCs which meet specific criteria. What these criteria are exactly isn't yet known, but I'm continuing investigating.