Kähler rigidity for manifolds with continuous symmetries

Let $(M_1,g_1,J_1,\omega_1)$ and $(M_2,g_2,J_2,\omega_2)$ be compact Kähler manifolds. Let $\operatorname{Isom_{\mathrm{Kah}}}(M_i)$ denote the Lie group of holomorphic isometries of $M_i$. Suppose there is a Lie-group isomorphism

$$\Phi:\operatorname{Isom_{\mathrm{Kah}}}(M_1)\cong\operatorname{Isom_{\mathrm{Kah}}}(M_2),$$

and both groups have positive dimension. Kähler rigidity conjecture. Then $M_1$ and $M_2$ are biholomorphic. Moreover, if $\Phi$ is induced by an algebraic correspondence between the groups of isometries, then $M_1$ and $M_2$ are isometric up to a constant scaling factor of their Kähler metrics. The conjecture concerns whether continuous global symmetry groups determine the underlying complex and Riemannian structures; the full statement remains open and is false without additional structure, such as curvature, stability, or special-metric assumptions.