Twisted K-stability and solvability conjecture

Let $(\Omega,\chi,t)$ be twisted Kähler data with $t<1$, and let the corresponding twisted continuity-path equation be the equation whose solvability at parameter $t$ defines a twisted cscK metric. Twisted K-stability and solvability conjecture. The triple $(\Omega,\chi,t)$ is twisted K-stable if and only if the corresponding equation admits a solution for the parameter $t$. The source defines twisted K-stability using the twisted Futaki invariant but does not establish this equivalence in general.