Closed immersions are preserved by the formal tangent scheme

Let $\operatorname{\mathfrak{X}}=(X_i)_{i\in I}$ be a formal scheme over a base scheme $S$, presented as an inductive system of schemes. Suppose that every morphism appearing in this formal scheme is a closed immersion. Closed-immersion preservation conjecture. Then every morphism appearing in the formal tangent scheme $T_{\operatorname{\mathfrak{X}}/S}$ is also a closed immersion. This is another proposed property of tangent categories for formal schemes; the supplied text gives no resolution status.