Twisted cscK existence and properness conjecture for generalized twisting forms

Let $\mu_k$ be the closed positive twisting data used to define the functional $J_{\mu_k}$, and let

$$E_{\mu_k,t}=tE+(1-t)J_{\mu_k}$$

be the corresponding twisted K-energy functional. A twisted cscK metric is a solution of the associated Euler–Lagrange equation. Twisted cscK properness conjecture. There exists a twisted cscK metric if and only if the corresponding twisted K-energy functional is proper in terms of geodesic distance. The source states this as the analogue of the properness criterion for the ordinary cscK problem and does not provide a general proof.