### 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
if (!docks_visible) {
		for (int i = 0; i < DOCK_SLOT_MAX; i++) {
			dock_slot[i]->hide();
		}
		for (int i = 0; i < vsplits.size(); i++) {
			vsplits[i]->hide();
		}
		right_hsplit->hide();
	} else {
		for (int i = 0; i < DOCK_SLOT_MAX; i++) {
			int first_tab_visible = -1;
			for (int j = 0; j < dock_slot[i]->get_tab_count(); j++) {
				if (!dock_slot[i]->is_tab_hidden(j)) {
					first_tab_visible = j;
					break;
				}
			}
			if (first_tab_visible >= 0) {
				dock_slot[i]->show();
				if (p_keep_selected_tabs) {
					int current_tab = dock_slot[i]->get_current_tab();
					if (dock_slot[i]->is_tab_hidden(current_tab)) {
						dock_slot[i]->set_block_signals(true);
						dock_slot[i]->select_next_available();
						dock_slot[i]->set_block_signals(false);
					}
				} else {
					dock_slot[i]->set_block_signals(true);