Uniqueness conjecture for semi-flat cscK currents on minimal elliptic surfaces

Let $\Phi:X\rightarrow\Sigma_g$ be a minimal elliptic surface, and let $H(X,\Phi)$ be the group of pairs of automorphisms preserving $\Phi$:

$$H(X,\Phi)=\{(h',h)\in\operatorname{Aut}(X)\times\operatorname{Aut}(\Sigma_g)\mid \Phi\circ h'=h\circ\Phi\}.$$

A semi-flat cscK current has fiber volume $1$ and scalar curvature $-3$. Uniqueness conjecture. There is a unique such current up to the action of $H(X,\Phi)$: for any two semi-flat cscK currents $\omega_1$ and $\omega_2$, there exists $(h',h)\in H(X,\Phi)$ such that

$$h'_*\omega_1=\omega_2.$$

This is the uniformization proposal stated after two further questions in the source. It overlaps with the preceding existence-and-uniqueness conjecture, but is phrased there as a separate uniformization suggestion; the source does not report a resolution.