Let $X_{t}$ be a family of compact complex manifolds over the disk $\mathbb{D}\subset \mathbb{C}.$ Formally, $X_{t}$ is the fiber over $t\in\mathbb{D}$ of a proper, holomorphic submersion of a complex manifold $\mathcal{X}$ onto $\mathbb{D}.$ Suppose $E_{t}$ is a family of holomorphic vector bundles over $X_{t}$ equipped with a family of holomorphic flat connections $\nabla^{t}.$
If $\mathbb{V}_{t}$ denotes the local system prescribed by $\nabla^{t},$ we have the inclusion of sheaves $0\rightarrow \mathbb{V}_{t}\rightarrow \mathcal{E}_{t}$ where $\mathcal{E}_{t}$ is the sheaf of germs of holomorphic sections of $E_{t}.$
Suppose that for all $t\neq 0$ in the disk, the induced map $H^{1}(X_{t}, \mathbb{V}_{t})\rightarrow H^{1}(X_{t}, \mathcal{E}_{t})$ is an isomorphism.
Question: Does it follow that the induced map over zero $H^{1}(X_{0}, \mathbb{V}_{0})\rightarrow H^{1}(X_{0}, \mathcal{E}_{0})$ is an isomorphism?
If it makes life any better, I am happy to assume that the dimension of $H^{1}(X_{t}, \mathbb{V}_{t})$ is constant over all of $\mathbb{D}.$
I suspect the answer is no, but haven't unearthed a counterexample as of yet. Thank you for any insight you can provide.