Wehler's weakened local-isomorphism conjecture for smooth families

Let $S$ be a complex manifold, and let $\pi:{\mathcal X}\to S$ and $\pi':{\mathcal Y}\to S$ be smooth families of compact complex manifolds. Write ${\mathcal X}_U=\pi^{-1}(U)$ and ${\mathcal Y}_U=\pi^{\prime-1}(U)$ for the restrictions over an open subset $U\subset S$. The families are pointwise isomorphic if, for every $s\in S$, there is a biholomorphism ${\mathcal Y}_s\to {\mathcal X}_s$; they are locally isomorphic at $s$ if some neighborhood of $s$ admits a biholomorphism ${\mathcal Y}_U\to {\mathcal X}_U$ over $U$. Wehler's weakened local-isomorphism conjecture. If the function

$$s\longmapsto \dim H^0({\mathcal Y}_s,T_{{\mathcal Y}_s})$$

is constant on $S$, then there exists an open dense subset $U\subset S$ such that the restricted families $\pi_U:{\mathcal X}_U\to U$ and $\pi'_U:{\mathcal Y}_U\to U$ are locally isomorphic at every point $s\in U$. This weakens Wehler's question, which asks for local isomorphism at every point of $S$; the result is relevant to the problem of comparing pointwise-isomorphic families of compact complex manifolds, and the source's abstract proves a related open-dense local-isomorphism result for projective non-uniruled manifolds.