### 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);
	return size_2d_override_stretch;
}
void SubViewport::set_update_mode(UpdateMode p_mode) {
	ERR_MAIN_THREAD_GUARD;
	update_mode = p_mode;
	RS::get_singleton()->viewport_set_update_mode(get_viewport_rid(), RS::ViewportUpdateMode(p_mode));
}
SubViewport::UpdateMode SubViewport::get_update_mode() const {
	ERR_READ_THREAD_GUARD_V(UPDATE_DISABLED);
	return update_mode;
}
void SubViewport::set_clear_mode(ClearMode p_mode) {
	ERR_MAIN_THREAD_GUARD;
	clear_mode = p_mode;
	RS::get_singleton()->viewport_set_clear_mode(get_viewport_rid(), RS::ViewportClearMode(p_mode));
}
SubViewport::ClearMode SubViewport::get_clear_mode() const {
	ERR_READ_THREAD_GUARD_V(CLEAR_MODE_ALWAYS);
	return clear_mode;
}
DisplayServer::WindowID SubViewport::get_window_id() const {
	ERR_READ_THREAD_GUARD_V(DisplayServer::INVALID_WINDOW_ID);
	return DisplayServer::INVALID_WINDOW_ID;
}
Transform2D SubViewport::get_screen_transform_internal(bool p_absolute_position) const {