### Godot version

4.0.3-stable

### System information

Windows 10

### Issue description

In some projects ViewportTexture remains pink and doesn't render viewport content after upgrading from 4.0.2-stable to 4.0.3-stable. If a pair of files viewport.h/viewport.cpp are downgraded back to what they were in 4.0.2-stable, problem goes away.

This is how it looks in 4.0.3-stable:
![4 0 3_stable_viewport_4 0 3](https://github.com/godotengine/godot/assets/1449921/44c191a9-4b67-4b02-90b6-4ab615f62a73)

And this is how it looks if viewport.h/viewport.cpp are downgraded back to what they were in 4.0.2-stable.
![4 0 3_stable_viewport_4 0 2](https://github.com/godotengine/godot/assets/1449921/f5abc605-f129-4222-9c0e-e9a07621dd9d)



### Steps to reproduce

Unfortunately, I cannot come up wit a simple demo project reproducing the problem. It exists in some projects and doesn't exist in others.


### Minimal reproduction project

Cannot produce a simple project reproducing the problem. In simple projects it works fine.
ERR_READ_THREAD_GUARD_V(false);
	if (!handle_input_locally) {
		ERR_FAIL_COND_V(!is_inside_tree(), false);
		const Viewport *vp = this;
		while (true) {
			if (Object::cast_to<Window>(vp)) {
				break;
			}
			if (!vp->get_parent()) {
				break;
			}
			vp = vp->get_parent()->get_viewport();
		}
		if (vp != this) {
			return vp->is_input_handled();
		}
	}
	return local_input_handled;
}
void Viewport::set_handle_input_locally(bool p_enable) {
	ERR_MAIN_THREAD_GUARD;
	handle_input_locally = p_enable;
}
bool Viewport::is_handling_input_locally() const {