Let $P_1, P_2$ and $P_3$ be Hilbert polynomials of projective schemes. Suppose that the flag Hilbert scheme $Hilb_{P_1,P_2}$ is non-empty. Assume further that there exists a closed immersion of the Hilbert scheme $Hilb_{P_3}$ to $Hilb_{P_2}$. Denote by $\mathcal{C}_1 \subset \mathcal{C}_2$ the universal family of the flag Hilbert scheme $Hilb_{P_1,P_2}$. Denote by $\mathcal{D}_1$ and $\mathcal{D}_2$ the universal families of $Hilb_{P_2}$ and $Hilb_{P_3}$, respectively.

Denote by $\mathcal{C}'_2$ the fiber product of $\mathcal{C}_2 \times_{\mathcal{D}_1} \mathcal{D}_2$.

Denote by $\mathcal{C}'_1$ the fiber product of $\mathcal{C}_1 \times_{\mathcal{C}_2} (\mathcal{C}_2 \times_{\mathcal{D}_1} \mathcal{D}_2)$.

Is it true that both $\mathcal{C}'_1$ and $\mathcal{C}'_2$ are flat over the fiber product $Hilb_{P_1,P_2} \times_{Hilb_{P_2}} Hilb_{P_3}$?

Let $P_1, P_2$ and $P_3$ be Hilbert polynomials of projective schemes. Suppose that the flag Hilbert scheme $Hilb_{P_1,P_2}$ is non-empty. Assume further that there exists a closed immersion of the Hilbert scheme $Hilb_{P_3}$ to $Hilb_{P_2}$. Denote by $\mathcal{C}_1 \subset \mathcal{C}_2$ the universal family of the flag Hilbert scheme $Hilb_{P_1,P_2}$. Denote by $\mathcal{D}_1$ and $\mathcal{D}_2$ the universal families of $Hilb_{P_2}$ and $Hilb_{P_3}$, respectively.

Denote by $\mathcal{C}'_2$ the fiber product of $\mathcal{C}_2 \times_{\mathcal{D}_1} \mathcal{D}_2$.

Is it true that $\mathcal{C}'_2$ is flat over the fiber product $Hilb_{P_1,P_2} \times_{Hilb_{P_2}} Hilb_{P_3}$?