### 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)

if (p_mode == EditorExportPreset::MODE_FILE_NOT_CUSTOMIZED) {
		p_item->set_checked(0, false);
		p_item->set_cell_mode(1, TreeItem::CELL_MODE_STRING);
		p_item->set_editable(1, false);
		p_item->set_selectable(1, false);
		p_item->set_custom_color(1, get_theme_color(SNAME("disabled_font_color"), EditorStringName(Editor)));
	} else {
		p_item->set_checked(0, true);
		p_item->set_cell_mode(1, TreeItem::CELL_MODE_CUSTOM);
		p_item->set_editable(1, true);
		p_item->set_selectable(1, true);
		p_item->clear_custom_color(1);
	}
	p_item->set_metadata(1, p_mode);
}
bool ProjectExportDialog::_fill_tree(EditorFileSystemDirectory *p_dir, TreeItem *p_item, Ref<EditorExportPreset> &current, EditorExportPreset::ExportFilter p_export_filter) {
	p_item->set_cell_mode(0, TreeItem::CELL_MODE_CHECK);
	p_item->set_icon(0, presets->get_theme_icon(SNAME("folder"), SNAME("FileDialog")));
	p_item->set_text(0, p_dir->get_name() + "/");
	p_item->set_editable(0, true);
	p_item->set_metadata(0, p_dir->get_path());
	if (p_export_filter == EditorExportPreset::EXPORT_CUSTOMIZED) {
		_setup_item_for_file_mode(p_item, current->get_file_export_mode(p_dir->get_path()));
	}
	bool used = false;
	for (int i = 0; i < p_dir->get_subdir_count(); i++) {
		TreeItem *subdir = include_files->create_item(p_item);
		if (_fill_tree(p_dir->get_subdir(i), subdir, current, p_export_filter)) {
			used = true;