In *Coherent analytic sheaves*, one has the following theorem due to Grauert:

Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V$ a holomorphic vector bundle on $X$. Then for any integers $q, d \geq 0$, the set $$ \left\{y \in Y: h^{q}\left(X_{y},\left.V\right|_{X_{y}}\right) \geq d\right\} $$ is an analytic subset of $Y$.

Hence, we can take $V=\Omega^p$ to get the hodge number. I wonder if the above assertion still holds for $(p,q)$-Bott–Chern cohomology? In other words, for the same conditions, is the set
$$
\left\{y \in Y: h^{p,q}_\text{BC}\left(X_{y}\right) \geq d\right\}
$$a complex analytic subset of $Y$?
Notice that one cannot apply Grauert's theorem to Bott–Chern cohomology, since now we just have the following isomorphism (one can refer to Demailly's book *Basic results on sheaves and analytic sets* for more details):
$$
H_\text{BC}^{p, q}(X) \cong \mathbb{H}^{p+q-1}\left(X, \mathscr{L}_{X}^{\bullet}\right).
$$

Basic results on sheaves and analytic sets. If that was not correct, please fix it to the correct reference. $\endgroup$