Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \mathbb{P}^n_B$, the composition of this closed immersion with the natural projection $\mathbb{P}^n_B \to B$ is $\pi$ and there exists a Hilbert polynomial $P$ such that for all $b \in B$, the fiber $\pi^{-1}(b)$ is of Hilbert polynomial $P$ in $\mathbb{P}^n_b$. Assume that $B$ is integral. Suppose that there exists a closed point $b_0 \in B$ such that the fiber $\pi^{-1}(b_0)$ can be embedded into $\mathbb{P}^3$ i.e., the closed immersion of $\pi^{-1}(b_0) \hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$. Is it then possible to find an open neighbourhood $U$ of $b_0$ such that for all $u \in U$, $\pi^{-1}(u)$ can be embedded into $\mathbb{P}^3_u$ i.e., the closed immersion of $\pi^{-1}(u) \hookrightarrow \mathbb{P}^n_u$ factors through $\mathbb{P}^3_u$? If not, is there any known condition on $B$ under which this happens?

Let $\pi:\mathcal{C} \to B$ be a (flat) family of complex projective schemes of pure dimension $1$ with fixed Hilbert polynomial, in particular, for some $n \ge 3$, $\mathcal{C} \hookrightarrow \mathbb{P}^n_B$, the composition of this closed immersion with the natural projection $\mathbb{P}^n_B \to B$ is $\pi$ and there exists a Hilbert polynomial $P$ such that for all $b \in B$, the fiber $\pi^{-1}(b)$ is of Hilbert polynomial $P$ in $\mathbb{P}^n_b$. Assume that $B$ is integral. Suppose that there exists a closed point $b_0 \in B$ such that the fiber $\pi^{-1}(b_0)$ can be embedded into $\mathbb{P}^3$ i.e., the closed immersion of $\pi^{-1}(b_0) \hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$. Is it then possible to find an open neighbourhood $U$ of $b_0$ such that for all $u \in U$, $\pi^{-1}(u)$ can be embedded into $\mathbb{P}^3_u$ i.e., the closed immersion of $\pi^{-1}(u) \hookrightarrow \mathbb{P}^n_u$ factors through $\mathbb{P}^3_u$? If not, is there any known condition on $B$ under which this happens?

EDIT By "closed immersion $i:A \hookrightarrow \mathbb{P}^n$ factors through $\mathbb{P}^3$", I mean that there is a closed immersion $i_1:A \hookrightarrow \mathbb{P}^3$ and a LINEAR embedding $i_2:\mathbb{P}^3 \hookrightarrow \mathbb{P}^n$ such that $i=i_2 \circ i_1$.