### 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)

bool has_messages = false;
	int msg_count = get_message_count();
	p_log->add_text(TTR("Project export for platform:") + " ");
	p_log->add_image(get_logo(), 16 * EDSCALE, 16 * EDSCALE, Color(1.0, 1.0, 1.0), INLINE_ALIGNMENT_CENTER);
	p_log->add_text(" ");
	p_log->add_text(get_name());
	p_log->add_text(" - ");
	if (p_err == OK) {
		if (get_worst_message_type() >= EditorExportPlatform::EXPORT_MESSAGE_WARNING) {
			p_log->add_image(p_log->get_editor_theme_icon(SNAME("StatusWarning")), 16 * EDSCALE, 16 * EDSCALE, Color(1.0, 1.0, 1.0), INLINE_ALIGNMENT_CENTER);
			p_log->add_text(" ");
			p_log->add_text(TTR("Completed with warnings."));
			has_messages = true;
		} else {
			p_log->add_image(p_log->get_editor_theme_icon(SNAME("StatusSuccess")), 16 * EDSCALE, 16 * EDSCALE, Color(1.0, 1.0, 1.0), INLINE_ALIGNMENT_CENTER);
			p_log->add_text(" ");
			p_log->add_text(TTR("Completed successfully."));
			if (msg_count > 0) {
				has_messages = true;
			}
		}
	} else {