### Tested versions

Godot Engine v4.2.stable.official.46dc27791

### System information

Windows 10

### Issue description

When exporting to web, Godot fails with this error in the console:

> scene/main/window.cpp:884 - Attempting to make child window exclusive, but the parent window already has another exclusive child. This window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->VBoxContainer@<!-- -->137/@<!-- -->AcceptDialog@<!-- -->411, parent window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559, current exclusive child window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->ProgressDialog@<!-- -->17444

![a](https://github.com/godotengine/godot/assets/2000051/bf9d22aa-d3fe-4e11-bff6-c387e740fe90)

This happens for newly created empty projects.


### Steps to reproduce

1. Open the attached project (or create a new project).
2. Project->Export->Export All
3. Observe the console error.  No output files are generated.


### Minimal reproduction project (MRP)

[export_test.zip](https://github.com/godotengine/godot/files/13574810/export_test.zip)

ep.step(TTR("Starting project..."), 5);
	err = ssh_run_on_remote_no_wait(host, port, extra_args_ssh, vformat("\"%s/%s\"", temp_dir, basepath.get_file() + "_start.sh"), &ssh_pid, (use_remote) ? dbg_port : -1);
	if (err != OK) {
		CLEANUP_AND_RETURN(err);
	}
	cleanup_commands.clear();
	cleanup_commands.push_back(SSHCleanupCommand(host, port, extra_args_ssh, vformat("\"%s/%s\"", temp_dir, basepath.get_file() + "_clean.sh")));
	print_line("Project started.");
	CLEANUP_AND_RETURN(OK);
#undef CLEANUP_AND_RETURN
}
EditorExportPlatformLinuxBSD::EditorExportPlatformLinuxBSD() {
	if (EditorNode::get_singleton()) {
#ifdef MODULE_SVG_ENABLED
		Ref<Image> img = memnew(Image);
		const bool upsample = !Math::is_equal_approx(Math::round(EDSCALE), EDSCALE);
		ImageLoaderSVG::create_image_from_string(img, _linuxbsd_logo_svg, EDSCALE, upsample, false);
		set_logo(ImageTexture::create_from_image(img));
		ImageLoaderSVG::create_image_from_string(img, _linuxbsd_run_icon_svg, EDSCALE, upsample, false);
		run_icon = ImageTexture::create_from_image(img);
#endif
		Ref<Theme> theme = EditorNode::get_singleton()->get_editor_theme();
		if (theme.is_valid()) {
			stop_icon = theme->get_icon(SNAME("Stop"), EditorStringName(EditorIcons));
		} else {
			stop_icon.instantiate();
		}
	}
}