Stability of extremal metrics under complex deformations

Let $(X,\Omega)$ be a closed polarized complex manifold, $g$ be an extremal metric on $X$ that represents the Kähler class $\Omega$, and $G$ be a compact connected subgroup of the isometry group $\mathrm{Isom}(X,g)$. Assume that the Futaki invariant relative to $G$ is nondegenerate at $g$. Consider a smooth family $(M\to B)$ of polarized complex deformations of $(X,\Omega)\simeq (M_0,\Theta_0)$ provided with a holomorphic action of $G$. Then for every $t\in B$ sufficiently small, there exists an $h^{1,1}(X)$-dimensional family of extremal Kähler metrics on $M_t$ whose Kähler classes are arbitrarily close to $\Theta_t$.