### 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 (selected_tab_idx >= 0 && selected_tab_idx < dock_slot[i]->get_tab_count()) {
			dock_slot[i]->call_deferred("set_current_tab", selected_tab_idx);
		}
	}
	for (int i = 0; i < vsplits.size(); i++) {
		if (!p_layout->has_section_key(p_section, "dock_split_" + itos(i + 1))) {
			continue;
		}
		int ofs = p_layout->get_value(p_section, "dock_split_" + itos(i + 1));
		vsplits[i]->set_split_offset(ofs);
	}
	for (int i = 0; i < hsplits.size(); i++) {
		if (!p_layout->has_section_key(p_section, "dock_hsplit_" + itos(i + 1))) {
			continue;
		}
		int ofs = p_layout->get_value(p_section, "dock_hsplit_" + itos(i + 1));
		hsplits[i]->set_split_offset(ofs);
	}
	_update_dock_slots_visibility(false);
	
	if (p_layout->has_section_key(p_section, "dock_filesystem_split")) {
		int fs_split_ofs = p_layout->get_value(p_section, "dock_filesystem_split");
		FileSystemDock::get_singleton()->set_split_offset(fs_split_ofs);
	}
	if (p_layout->has_section_key(p_section, "dock_filesystem_display_mode")) {
		FileSystemDock::DisplayMode dock_filesystem_display_mode = FileSystemDock::DisplayMode(int(p_layout->get_value(p_section, "dock_filesystem_display_mode")));
		FileSystemDock::get_singleton()->set_display_mode(dock_filesystem_display_mode);
	}
	if (p_layout->has_section_key(p_section, "dock_filesystem_file_sort")) {
		FileSystemDock::FileSortOption dock_filesystem_file_sort = FileSystemDock::FileSortOption(int(p_layout->get_value(p_section, "dock_filesystem_file_sort")));