Kawakami–Totaro conjecture on endomorphisms of Fano varieties of Picard number 1

Let $X$ be a smooth Fano variety of Picard number $\rho(X)=1$ over an algebraically closed field of characteristic zero. An endomorphism of degree greater than $1$ is a self-map of $X$ whose degree is greater than $1$. Kawakami–Totaro conjecture. If $X$ admits an endomorphism of degree greater than $1$, then

$$X\simeq \mathbb P^n.$$

The conjecture predicts that projective space is the only smooth Fano variety of Picard number $1$ admitting a self-map of degree greater than $1$. The supplied text does not state whether the conjecture is resolved.