Completeness of fiber products of complete transverse exploded morphisms

Let $f$ and $g$ be complete and transverse smooth exploded morphisms

$$\mathfrak A\xrightarrow{f}\mathfrak C\xleftarrow{g}\mathfrak B.$$

The fiber product $\mathfrak A{}_{\hspace{3pt}f\hspace{-2pt}}\times_{g}\mathfrak B$ exists, and its natural morphism to $\mathfrak C$ is complete:

$$\mathfrak A{}_{\hspace{3pt}f\hspace{-2pt}}\times_{g}\mathfrak B\longrightarrow\mathfrak C.$$

The statement concerns stability of completeness under transverse fiber products; the existence of these fiber products is described earlier and is noted there as an exercise, with some special cases known. No proof or resolution of this completeness claim is supplied here.