### 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)

name->set_editable(true);
	export_path->show();
	duplicate_preset->set_disabled(false);
	delete_preset->set_disabled(false);
	get_ok_button()->set_disabled(false);
	name->set_text(current->get_name());
	List<String> extension_list = current->get_platform()->get_binary_extensions(current);
	Vector<String> extension_vector;
	for (int i = 0; i < extension_list.size(); i++) {
		extension_vector.push_back("*." + extension_list[i]);
	}
	export_path->setup(extension_vector, false, true);
	export_path->update_property();
	runnable->set_disabled(false);
	runnable->set_pressed(current->is_runnable());
	parameters->set_object_class(current->get_platform()->get_class_name());
	parameters->edit(current.ptr());
	export_filter->select(current->get_export_filter());
	include_filters->set_text(current->get_include_filter());
	include_label->set_text(_get_resource_export_header(current->get_export_filter()));
	exclude_filters->set_text(current->get_exclude_filter());
	server_strip_message->set_visible(current->get_export_filter() == EditorExportPreset::EXPORT_CUSTOMIZED);
	_fill_resource_tree();
	bool needs_templates;
	String error;
	if (!current->get_platform()->can_export(current, error, needs_templates)) {
		if (!error.is_empty()) {
			Vector<String> items = error.split("\n", false);
			error = "";
			for (int i = 0; i < items.size(); i++) {