Uniqueness conjecture for the Moishezon element of the twistor-group action

Let $\Gamma_{\mathcal X}$ be the group of elements defining the corresponding $\gamma$-twistor spaces, and call an element $\gamma\in\Gamma_{\mathcal X}$ Moishezon when the algebraic dimension of its twistor space satisfies

$$a(\mathrm{TW}^{\gamma}(X,r))=1+\dim_{\mathbb C}M_{\mathrm{dR}}(X,r).$$

Moishezon-element uniqueness conjecture. There is a unique Moishezon element in $\Gamma_{\mathcal X}$, namely the equivalence class of complex conjugation. The conjecture asserts that among the twistor spaces parametrized by $\Gamma_{\mathcal X}$, exactly one equivalence class has the maximal algebraic dimension specified above; its resolution is not given in the source.