### 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
Control *right_spacer = memnew(Control);
	right_spacer->set_mouse_filter(Control::MOUSE_FILTER_PASS);
	right_spacer->set_h_size_flags(Control::SIZE_EXPAND_FILL);
	title_bar->add_child(right_spacer);
	project_run_bar = memnew(EditorRunBar);
	title_bar->add_child(project_run_bar);
	project_run_bar->connect("play_pressed", callable_mp(this, &EditorNode::_project_run_started));
	project_run_bar->connect("stop_pressed", callable_mp(this, &EditorNode::_project_run_stopped));
	HBoxContainer *right_menu_hb = memnew(HBoxContainer);
	title_bar->add_child(right_menu_hb);
	renderer = memnew(OptionButton);
	renderer->set_visible(true);
	renderer->set_flat(true);
	renderer->set_fit_to_longest_item(false);
	renderer->set_focus_mode(Control::FOCUS_NONE);
	renderer->add_theme_font_override("font", theme->get_font(SNAME("bold"), EditorStringName(EditorFonts)));
	renderer->add_theme_font_size_override("font_size", theme->get_font_size(SNAME("bold_size"), EditorStringName(EditorFonts)));
	renderer->set_tooltip_text(TTR("Choose a renderer."));
	right_menu_hb->add_child(renderer);
	if (can_expand) {
		
		right_menu_spacer = memnew(Control);
		right_menu_spacer->set_mouse_filter(Control::MOUSE_FILTER_PASS);
		title_bar->add_child(right_menu_spacer);
	}
	String current_renderer_ps = GLOBAL_GET("rendering/renderer/rendering_method");
	current_renderer_ps = current_renderer_ps.to_lower();
	String current_renderer_os = OS::get_singleton()->get_current_rendering_method().to_lower();