Carlo Rovelli will publicly admit that the claim is problematic.

Francesca Vidotto will publicly admit that the claim is problematic.

Ding Jia (question creator) will publicly admit that his own criticism is problematic.

The claim will be modified or removed in a new edition of the textbook.

In their influential textbook "Covariant loop quantum gravity" physicists Rovelli and Vidotto claim that: "The formulation of the gravitational field as a pseudo-Riemannian metric cannot be fundamentally correct, because it does not allow coupling to fermions, and fermions exist in the world. For this, we need the tetrad formulation."

I think this claim is wrong, for reasons summarized in this youtube video: 

This market is for proposition bets regarding this issue.

https://manifold.markets/ttoe/is-david-wallace-wrong-about-decohe

https://manifold.markets/ttoe/are-feldbrugge-lehners-turok-wrong