Existence and uniqueness conjecture for semi-flat cscK currents on minimal elliptic surfaces

Let $\Phi:X\rightarrow\Sigma_g$ be a minimal elliptic surface. A semi-flat cscK current is a semi-flat constant scalar curvature Kähler current on $X$. Let

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

Existence and uniqueness conjecture. There is a semi-flat cscK current on $X$ whose regular fibers have volume $1$ and whose scalar curvature is $-3$; moreover, if $\omega_1$ and $\omega_2$ are any two such currents, then there exists $(h',h)\in H(X,\Phi)$ such that

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

The conjecture asserts existence and uniqueness up to automorphisms preserving the elliptic fibration. The source presents it as a proposed conjecture and gives no resolution.