Delta-invariant conjecture for the exceptional index-two family

Let $S_n$ be a quasi-smooth del Pezzo hypersurface with weighted-projective quintuple

$$(a_0,a_1,a_2,a_3,d)=(1,1,n+1,n+1,2n+2).$$

Delta-invariant conjecture. For every $n$, one has

$$\delta(S_n)=1,$$

so $S_n$ is K-semistable; moreover, $S_n$ admits an orbifold Kähler–Einstein metric.

The source notes that the delta invariant is already known to satisfy $\delta(S_n)\leq 1$ for this family and presents equality, together with metric existence, as the expected conclusion. The claim remains open in the supplied material.