### 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
load_scene(open_import_request, true, false, true, true);
}
void EditorNode::dim_editor(bool p_dimming) {
	dimmed = p_dimming;
	gui_base->set_modulate(p_dimming ? Color(0.5, 0.5, 0.5) : Color(1, 1, 1));
}
bool EditorNode::is_editor_dimmed() const {
	return dimmed;
}
void EditorNode::open_export_template_manager() {
	export_template_manager->popup_manager();
}
void EditorNode::add_resource_conversion_plugin(const Ref<EditorResourceConversionPlugin> &p_plugin) {
	resource_conversion_plugins.push_back(p_plugin);
}
void EditorNode::remove_resource_conversion_plugin(const Ref<EditorResourceConversionPlugin> &p_plugin) {
	resource_conversion_plugins.erase(p_plugin);
}
Vector<Ref<EditorResourceConversionPlugin>> EditorNode::find_resource_conversion_plugin(const Ref<Resource> &p_for_resource) {
	Vector<Ref<EditorResourceConversionPlugin>> ret;
	for (int i = 0; i < resource_conversion_plugins.size(); i++) {
		if (resource_conversion_plugins[i].is_valid() && resource_conversion_plugins[i]->handles(p_for_resource)) {
			ret.push_back(resource_conversion_plugins[i]);
		}
	}
	return ret;
}
void EditorNode::_bottom_panel_raise_toggled(bool p_pressed) {