Tested versions

Reproducible in Godot 4.4.dev - dev7
Not Reproducible in Godot 4.4.dev3

System information

Godot v4.4.dev7 - Linux Mint 22 (Wilma) on X11 - X11 display driver, Multi-window, 2 monitors - OpenGL 3 (Compatibility) - NVIDIA GeForce RTX 4060 Ti (nvidia; 550.120) - AMD Ryzen 9 3900X 12-Core Processor (24 threads)

Issue description

As of Godot 4.4.dev4 I am unable to connect to Web export projects to see the output, monitors, or the remote tree.
I expected it to connect to the web export as was possible in earlier versions.

Steps to reproduce

Use Godot 4.4.dev3 and launch the web export with Debug -> Deploy With Remote Debug and Debug -> Keep Remote Server Open enabled. Notice the web export project connects back to your editor so you can more easily debug any problems.
Now launch the same project as a web export in Godot 4.4.dev4 and notice it never connects to the editor.

Minimal reproduction project (MRP)

WebDebugOutputBroken.zip