K-stability conjecture for the remaining index-two del Pezzo hypersurfaces

Let $S_d$ be a singular del Pezzo hypersurface with index $I=2$, and let $(a_0,a_1,a_2,a_3,d)$ denote its weighted-projective quintuple. Exclude the five exceptional quintuples listed in the preceding theorem and also exclude

$$(1,1,n+1,n+1,2n+2).$$

K-stability conjecture. Then $S_d$ is $K$-stable, and hence admits an orbifold Kähler–Einstein metric.

This is a refinement of the general index-two existence claim after isolating the five families proved not to admit orbifold Kähler–Einstein metrics and the separate family whose delta invariant is treated independently. The statement is presented as a conjecture in the source; no resolution is supplied in the given material.