Vanishing of the top log Bando–Futaki invariant for a momentum-constructed conical higher cscK metric

Let $X$) be the minimal ruled surface considered in the paper, with distinguished sections $S_0$ and $S_\infty$, and let $\omega$ be a conical higher cscK metric on $X$ with cone angles $2\pi\beta_0$ and $2\pi\beta_\infty$ along $S_0$ and $S_\infty$, respectively. Assume that $\omega$ belongs to the Kähler class

$$2\pi\left(\mathsf{C}+mS_\infty\right)$$

and is yielded by the momentum construction described in the paper. Let $\eta$ be any smooth Kähler metric in the same Kähler class, and let $w\frac{\partial}{\partial w}$ be the relevant holomorphic vector field. Vanishing of the top log Bando–Futaki invariant. One should have

$$\mathcal{F}_{\log;\,\beta_0,\beta_\infty}\left(w\frac{\partial}{\partial w},\eta\right)=\mathcal{F}_{\log;\,\beta_0,\beta_\infty}\left(w\frac{\partial}{\partial w},\omega\right)=0.$$

The claim predicts that, in this momentum-constructed conical higher cscK setting, the top log Bando–Futaki invariant is independent of the chosen metric in the Kähler class and vanishes. The source gives no proof and explicitly presents this as a belief motivated by the expected vanishing of the common value; its status is therefore open.