Log Donaldson–Tian–Yau conjecture

Let $X$ be a projective variety, let $D$ be a divisor on $X$, let $L$ be a polarisation, and let $\beta$ be a cone-angle parameter. The pair $((X,D);L)$ is log $K$-polystable with cone angle $2\beta\pi$ when it is log $K$-semistable and the log Donaldson–Futaki invariant vanishes only for product log test configurations. A cscK metric in $c_1(L)$ with cone singularities along $D$ has cone angle $2\pi\beta$ along $D$.

Log Donaldson–Tian–Yau conjecture. $((X,D);L)$ is log $K$-polystable with cone angle $2\pi\beta$ if and only if $X$ admits a cscK metric in $c_1(L)$ with cone singularities along $D$ with cone angle $2\pi\beta$.

This is presented as a folklore conjecture extending the Donaldson–Tian–Yau correspondence to pairs with cone singularities. The source gives supporting evidence in momentum-constructed examples, but the general statement remains open.