### Godot version

4.2

### System information

Linux

### Issue description

A nested state machine isn't traveling to its animation in `_ready()` in 4.2 as it was in 4.1.3.

In the ready function, we make the `root` state machine travel to the state of the `Stages` nested state machine. Then we immediately call the `Stages` nested state machine to travel to `Stage2`. This works fine in Godot v4.1.3, but not in Godot v4.2, where it seems a delay is first needed for the second travel animation to work.

![image](https://github.com/godotengine/godot/assets/788630/f1582a99-a68e-45d9-82b8-8367e62a9cba)
![image](https://github.com/godotengine/godot/assets/788630/c7c1f9fa-c78b-4259-bd36-81ebd8cef9aa)

### Steps to reproduce

Run the MRP below in 4.1.3 and 4.2.

In 4.1.3, the animation tree will reach `Stage2` and show Finished on the screen.

In 4.2, the animation tree will only reach `Stage1` and show Stuck on the screen.

### Minimal reproduction project

[MRP-AnimationNodeStateMachinePlayback.zip](https://github.com/godotengine/godot/files/13533132/MRP-AnimationNodeStateMachinePlayback.zip)

ERR_FAIL_COND(!property_reference_map.has(p_oid));
	String base_path = String(property_reference_map[p_oid]) + String(p_node);
	for (const PropertyInfo &E : properties) {
		if (E.name.begins_with(base_path)) {
			property_map.erase(E.name);
		}
	}
	// 
	properties_dirty = true;
	_update_properties();
}
void AnimationTree::_update_properties_for_node(const String &p_base_path, Ref<AnimationNode> p_node) {
	ERR_FAIL_COND(p_node.is_null());
	if (!property_parent_map.has(p_base_path)) {
		property_parent_map[p_base_path] = HashMap<StringName, StringName>();
	}
	if (!property_reference_map.has(p_node->get_instance_id())) {
		property_reference_map[p_node->get_instance_id()] = p_base_path;
	}
	if (p_node->get_input_count() && !input_activity_map.has(p_base_path)) {
		Vector<Activity> activity;