On a fresh Firefox 144.0 and Heliosphere 4.4.4, I can connect fine using that page. Something is blocking your requests. Does it still happen in Firefox troubleshoot mode?
If the issue goes away in troubleshoot mode, an extension is to blame. If you find out which extension, please let me know.
If troubleshoot mode does not fix it, I would also check anti-virus if you're on Windows.

/hs, click on the Settings tab, then scroll down to the Miscellaneous section. Expand that section and see if there's a button that says "Try starting server". If that buttons appears, something prevented the server from starting. In which case, please send your dalamud.log file.I cannot reproduce this issue; it is likely something local on your machine. Check firewalls and other "security" features that block localhost.
In-game you can try/hs, click on the Settings tab, then scroll down to the Miscellaneous section. Expand that section and see if there's a button that says "Try starting server". If that buttons appears, something prevented the server from starting. In which case, please send your dalamud.log file.
This is relevant from another thread.Oh, wow, I think it may be a Windows 11 update causing issues. See this thread on StackOverflow. It works fine for me on Fedora, which is probably why I can't reproduce this issue!