### 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)

include_files->clear();
	include_label->hide();
	include_margin->hide();
	Ref<EditorExportPreset> current = get_current_preset();
	if (current.is_null()) {
		return;
	}
	EditorExportPreset::ExportFilter f = current->get_export_filter();
	if (f == EditorExportPreset::EXPORT_ALL_RESOURCES) {
		return;
	}
	TreeItem *root = include_files->create_item();
	if (f == EditorExportPreset::EXPORT_CUSTOMIZED) {
		include_files->set_columns(2);
		include_files->set_column_expand(1, false);
		include_files->set_column_custom_minimum_width(1, 250 * EDSCALE);
	} else {
		include_files->set_columns(1);
	}
	include_label->show();
	include_margin->show();
	_fill_tree(EditorFileSystem::get_singleton()->get_filesystem(), root, current, f);
	if (f == EditorExportPreset::EXPORT_CUSTOMIZED) {
		_propagate_file_export_mode(include_files->get_root(), EditorExportPreset::MODE_FILE_NOT_CUSTOMIZED);
	}
}
void ProjectExportDialog::_setup_item_for_file_mode(TreeItem *p_item, EditorExportPreset::FileExportMode p_mode) {