### 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
internal_margin = memnew(MarginContainer);
	add_child(internal_margin, false, INTERNAL_MODE_FRONT);
	VBoxContainer *real_vbox = memnew(VBoxContainer);
	internal_margin->add_child(real_vbox);
	HBoxContainer *hb_edit = memnew(HBoxContainer);
	real_vbox->add_child(hb_edit);
	hb_edit->set_v_size_flags(SIZE_SHRINK_BEGIN);
	uv_edit = memnew(Control);
	hb_edit->add_child(uv_edit);
	uv_edit->connect("gui_input", callable_mp(this, &ColorPicker::_uv_input).bind(uv_edit));
	uv_edit->set_mouse_filter(MOUSE_FILTER_PASS);
	uv_edit->set_h_size_flags(SIZE_EXPAND_FILL);
	uv_edit->set_v_size_flags(SIZE_EXPAND_FILL);
	uv_edit->connect("draw", callable_mp(this, &ColorPicker::_hsv_draw).bind(0, uv_edit));
	sample_hbc = memnew(HBoxContainer);
	real_vbox->add_child(sample_hbc);
	btn_pick = memnew(Button);
	sample_hbc->add_child(btn_pick);
	if (DisplayServer::get_singleton()->has_feature(DisplayServer::FEATURE_SCREEN_CAPTURE)) {
		btn_pick->set_tooltip_text(RTR("Pick a color from the screen."));
		btn_pick->connect(SNAME("pressed"), callable_mp(this, &ColorPicker::_pick_button_pressed));
	} else {
		
		btn_pick->set_tooltip_text(RTR("Pick a color from the application window."));
		btn_pick->connect(SNAME("pressed"), callable_mp(this, &ColorPicker::_pick_button_pressed_legacy));
	}