This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a complex manifold $\mathcal{X}$ together with a proper holomorphic submersion $\pi: \mathcal{X}\to \Delta$ such that $\pi^{-1}(0)$ is biholomorphically isomorphic to $X$. Notice that in general the fiber $\pi^{-1}(t)$ is $C^{\infty}$-isomorphic to $X$ but not biholomorphically.
Two families of deformations of $X$, $\pi: \mathcal{X}\to \Delta$ and $\tilde{\pi}:\tilde{\mathcal{X}} \to \Delta$ are called equivalent if there exists a biholomorphic isomorphism $\phi: \mathcal{X}\overset{\sim}{\to} \tilde{\mathcal{X}}$ such that $\phi\circ \tilde{\pi}=\pi$ and $\phi|_{\pi^{-1}(0)}=\text{id}_X$.
Now we can loose the condition: Consider a $C^{\infty}$-isomorphism $\phi: \mathcal{X}\overset{\sim}{\to} \tilde{\mathcal{X}}$ such that
- $\phi\circ \tilde{\pi}=\pi$.
- $\phi$ is a biholomorphic isomorphism when restricted to a fiber of $\pi$ at any $t\in \Delta$.
- $\phi|_{\pi^{-1}(0)}=\text{id}_X$.
We call such $\phi$ a $C^{\infty}$-equivalence between $\mathcal{X}$ and $\tilde{\mathcal{X}}$.
My question is: when $\Delta$ is small enough, if there exists a $C^{\infty}$-equivalence between two families of deformations of $X$, is it always true that there exists a equivalence between them?
Notice that if $\tilde{\pi}:\tilde{\mathcal{X}} \to \Delta$ is the trivial deformation, i.e. $\tilde{\mathcal{X}}=X\times \Delta$ and $\tilde{\pi}$ is the projection to the second component, then the above result is implied by the Grauert-Fischer theorem:
"For a family of compact complex manifold $\mathcal{X}$ over $\Delta$, if each fiber $X_t$ is biholomorphically isomorphic to $X_0$, then $\mathcal{X}$ is biholomorphically isomorphic to $X_0\times \Delta$."
But I am not sure if the similar statement is true for non-trivial families. By the way, I don't have the access to Grauert-Fischer's original paper Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten so I cannot see if their proof could be generalized to this case.
