Let $\pi \colon \mathfrak{X} \rightarrow B$ be a deformation of complex compact manifolds and $E$ be a holomorphic vector bundle on $\mathfrak{X}$ (or a coherent sheaf on $\mathfrak{X}$ that is flat over $B$). The function $b \rightarrow \mathrm{h}^0(X_b, E_{|X_b})$ is known to be upper semi-continuous, but I wonder if it is constructible.

If the deformation is algebraic, that is if there exists a relatively ample line bundle on $\mathfrak{X}$, then there exists a complex $\mathcal{L}^{\bullet}$ of locally free sheaves on ${B}$ such that for any $b$ in $B$ and any nonnegative integer $i$, $$ \mathrm{H}^i(\mathcal{L}^{\bullet}_{|b}) \simeq \mathrm{H}^i(X_b, E_{| X_b}). $$ This is proved for instance in [Voisin, Complex algebraic geometry, Vol. 1, section 9.3.1], and implies constructibility.

Is the same result known in the complex analytic case?