The anticanonical boundary conjecture for klt Fano varieties

Let $X$ be a klt Fano variety of dimension $n$ with stability threshold $\delta(X)<1$. A general member $D$ of the rational anticanonical linear system $|-K_X|_{\mathbb Q}$ defines a pair with boundary coefficient $1-\delta(X)$.

Anticanonical Boundary Conjecture. Suppose $X$ is a klt Fano variety of dimension $n$ with $\delta(X)<1$. Then there is a general $D\in |-K_X|_{\mathbb Q}$ such that $(X,(1-\delta(X))D)$ is a klt log Fano pair which is K-semistable and satisfies

$$\delta(X,(1-\delta(X))D)=1.$$

The supplied passage is inside a commented-out discussion and gives no evidence of resolution. The claim predicts a K-semistable log Fano boundary obtained from a general anticanonical divisor.