### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
List<StringName> tl;
	theme->get_icon_list(EditorStringName(EditorIcons), &tl);
	for (const StringName &E : tl) {
		if (!ClassDB::class_exists(E)) {
			continue;
		}
		icon_type_cache[E] = theme->get_icon(E, EditorStringName(EditorIcons));
	}
}
void EditorNode::_enable_pending_addons() {
	for (uint32_t i = 0; i < pending_addons.size(); i++) {
		set_addon_plugin_enabled(pending_addons[i], true);
	}
	pending_addons.clear();
}
void EditorNode::_file_dialog_register(FileDialog *p_dialog) {
	singleton->file_dialogs.insert(p_dialog);
}
void EditorNode::_file_dialog_unregister(FileDialog *p_dialog) {
	singleton->file_dialogs.erase(p_dialog);
}
void EditorNode::_editor_file_dialog_register(EditorFileDialog *p_dialog) {